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Direction

Direction

The term direction can be applied to various topics.
- For the management, supervision, or guidance of a film, see film director
- For the guidance and cueing of a group of musicians during performance, see conducting
- A complex number x on the unit circle.

x\bar x=1

- A direction can be specified by a unit vector; the components are direction cosines; together with magnitude, direction can be specified by a vector in general
- For determination of position, see navigation, compass or left and right
- For finding the direction to a radio source, see radio direction finder
- Direction (Alexander Technique), a principal tool used in practicing the Alexander Technique
- Direction - Social Democracy, a major political party in Slovakia simple:Direction

Film director

A film director orchestrates the artistic and dramatic aspects of a film. The role typically includes:
- Defining the overall artistic vision of the film.
- Controlling the content and flow of the film's plot.
- Directing the performances of actors, both mechanically by putting them in certain positions (i.e. blocking), and dramatically by eliciting the required range of emotions.
- Organizing and selecting the locations in which the film will be shot.
- Managing technical details such as the positioning of cameras, the use of lighting, and the timing and content of the film's soundtrack.
- Any other activity that defines or realizes the artistic vision the director has for the film. In practice the director will delegate many of these responsibilities to other members of his or her film crew. For example, the director may describe the mood she or he wants from a scene, then leave it to other members of the film crew to find a suitable location, or to set up the appropriate lighting. The degree of control that a director exerts over a film varies greatly. Many directors are usually, but not essentially, subordinate to the studio and producer. This was especially true during the "Golden Era" of Hollywood from the 1930s through the 1950s, when studios had stables of directors, actors and writers under contract. Other directors bring a particular and intensely focused artistic vision to the pictures they make (see auteur theory). Their methods range from some who like to outline a general plot line and let the actors improvise dialogue (such as Robert Altman and Christopher Guest), to those who control every aspect, and demand that the actors and crew follow instructions precisely (such as Alfred Hitchcock, Charlie Chaplin, and Stanley Kubrick). Some directors also write their own scripts (such as James Cameron, Frank Darabont, and Quentin Tarantino), while others collaborate on screenplays with long-standing writing partners (such as Billy Wilder and his writing partner I.A.L. Diamond). Finally, certain directors star, often in leading roles, in their films, from Orson Welles to Woody Allen and Clint Eastwood to Mel Brooks. Directors often work closely with film producers, who are usually responsible for the non-artistic elements of the film, such as financing, contract negotiation and marketing. Directors will often take on some of the responsibilities of the producer for their films (e.g. Steven Spielberg), or work so closely with the producer that the distinction in their roles becomes blurred (as is the case with Joel and Ethan Coen). The early silent film director Alice Guy Blaché not only produced her own pictures but actually created her own highly successful studio. The official American film directors' trade union is the Directors Guild of America (DGA). In DGA pictures the credit for the director will always be the last credit in the film's title sequence. Directors, however, often get a second credit, "An (Insert Director Here) Film". The SAG has attacked this credit during contract negotations, arguing that it implies that directors have more authorship of films than actors. The key person in the making of a film is the director, the individual who visualizes the script and guides the production crew and actors to carry out that vision. The director has artistic control over everything from the script itself to the final cut of the film. It is the director's sense of the dramatic along with the creative visualization of the script that transforms a story into a well-made motion picture. The director is usually selected by the producer. Along with the producer, the director then puts together the production team

External links

- [http://www.medialawyer.com/DIRART2.htm Smooth Negotiating: Making the Director Deal]
- [http://www.dga.org Director's Guild of America] Category:Entertainment occupations
- Film director Category:Film crew ja:映画監督

Conducting

See Conductor for other possible uses of the word. ---- Conductor Conducting is the act of directing a musical performance by way of visible gestures. Orchestras, choirs and other musical ensembles often have conductors. A conductor resident with an orchestra (as opposed to a guest conductor) who has involvement with the artistic direction of an orchestra or opera company is sometimes known as a musical director, or nowadays by the German word Kapellmeister. Choir conductors are sometimes called choral directors. Respected senior conductors (like senior instrumentalists) are sometimes referred to by the Italian word Maestro ("master").

History of conducting

An early form of conducting is cheironomy, the use of hand gestures to indicate melodic shape. This has been practiced at least as far back as the middle ages. In the Christian church, the person giving these symbols held a staff to signify his role, and it seems that as music became more rhythmically involved, the staff was moved up and down to indicate the beat, acting as an early form of baton. From around the 17th century other devices to indicate the passing of time were used. Rolled up sheets of paper, smaller sticks and unadorned hands are all shown being used in contemporary pictures. The large staff remained in use at the Paris Opera, and was responsible for the death of Jean-Baptiste Lully - he hit his foot with the staff while conducting, and the wound became gangrenous. In instrumental music, a single performer usually acted as the conductor. This could be the principal violinist, who used his bow as a baton, or a lutenist who would move the neck of his instrument in time with the beat. It was also common to conduct from the harpsichord in pieces which had a basso continuo part. In opera performances there were sometimes two conductors - one at the keyboard in charge of the singers, and the principal violinist in charge of the orchestra. By the early 19th century, music had become sufficiently complex that it was desirable to have one person dedicated to conducting, not having to concern himself with performing as well. Accordingly, the baton became more common - this had the added advantage of being easier to see than bare hands or rolled-up paper by the orchestra, which was at this time expanding in size. Among the earliest notable conductors were Louis Spohr, Carl Maria von Weber and Felix Mendelssohn, all of them also composers. Felix Mendelssohn Hector Berlioz and Richard Wagner were also conductors, and they wrote two of the earliest essays dedicated to the subject. Wagner was largely responsible for shaping the conductor's role as somebody who imposes his own view of a piece onto the performance rather than somebody who is simply responsible for ensuring entries are made at the right time and that there is a unified beat.

Conducting technique

Since conducting is essentially a means of communicating 'real-time' instructions from the conductor to the performers, the only golden rule of conducting technique is that it should be clear and easy to follow. Aside from this, there are no hard-and-fast rules on how to conduct 'correctly', and a wide variety of different conducting styles exists. An understanding of the three basic elements of musical expression (tempo, dynamics and style) makes any conductor capable of conveying a basic interpretation of the music to the performers. However, it does not give the conductor mastery of the nuances of shading and phrasing that can only come with experience. There is a particular distinction between orchestral conducting and choral conducting. Orchestral conductors typically use a baton (though not always: this is up to the conductor's personal preference), and giving a clear beat to the players is central. Choral conductors rarely use a baton, and although the beat is an important part of choral conducting, conductors tend to concentrate on musical expression and shape, making their movements appear more abstract. Despite this wide variety of styles, a number of standard conventions have developed.

Beat and tempo

The beat of the music is typically indicated with the conductor's right hand, with or without a baton. The hand traces a shape in the air in every bar (measure) depending on the time signature. For music in simple quadruple time (four beats in a bar), the hand traces down-left-right-up. For music in simple triple time (three beats in a bar), the hand traces down-right-up. For music in simple duple time (two beats in a bar), the hand or baton traces down-up or down&right-up. The two most important movements are the downbeat, which indicates the first beat of the bar, and the upbeat, which indicates the last beat of the bar. The instant at which the beat occurs is called the ictus (plural: ictus or ictuses), usually indicated by a sudden (though not necessarily large) change in hand (or baton) motion. The gesture leading up to the ictus is called the preparation, and the conductor's principal responsibility is to provide a preparation which forecasts with certainty the exact moment of the coming ictus, so that all the players (or singers) can play simultaneously. If the tempo is slow or slowing, or if the time signature is compound, a conductor will sometimes indicate 'subdivisions' of the beats. For instance, in a particularly slow quadruple time, the conductor may beat down-and-left-and-right-and-up-and, where each 'and' is marked with a movement to an intervening point in the shape that is traced in the air. Some conductors use both hands to indicate the beat, with the hands to mirror each other's movements, though others view this as redundant and therefore to be avoided. In any case, the second hand is also used for turning pages in the sheet music, cueing the entrances of individual players or sections, and indicating other aspects of expression.

Tempo

Changes to the speed of the music are indicated simply by changing the speed of the beat. To encourage a particular accelerando or rallentando, a conductor may use additional body language such as leaning forward or back, increasing eye contact, making circling motions with the hands, or introducing beat subdivisions.

Dynamics

Dynamics are indicated in two main ways. Firstly, the volume of the music can be communicated via the size of the conducting movements: the larger the shape, the louder the sound. Secondly, changes to volume can be signalled with the hand that is not being used to indicate the beat: an upward motion (usually palm-up) indicates a crescendo, a downward motion (usually palm-down) indicates a diminuendo. The former, changing the size of movements, often results in unintended tempo changes as well, that is, larger movements tend to slow down the tempo. Therefore, many conductors also change the tension of the hands, whereby the required change in size of movements is smaller. Loud dynamics would then correspond to strained muscles and rigid movements, while soft dynamics correspond to relaxed hands and soft movements. Volume can be fine-tuned using various intuitive signals: for instance, showing one's palm to the performers in a 'stop' gesture, leaning away from them or putting a finger to the lips can be used to demonstrate a decrease in volume. In choral conducting, wiggling the fingers of the right hand is also an accepted signal for 'sing much more quietly'. All these signals can be combined with eye contact or pointing to particular sections or performers in order to adjust the overall balance of the various instruments or voices.

Entries

Another important task for the conductor is indicating 'entries', i.e. moments when a new instrument or section joins the music. Such indications are often called 'cues', and the process of giving such indications is called 'cueing'. Cueing is done either by pointing at the section at the appropriate time (though many orchestral players consider this poor etiquette) or by eye contact. In the case of complex music where several parts enter simultaneously, the latter is obviously more practical. When more than one section of the ensemble enters at the same time, it is often effective simply to look in their general direction. While most entries only require eye contact, a bigger moment in the music may warrant the use of a hand gesture. This could be deliberately choreographed beforehand by the conductor when studying the score, or a spontaneous gesture designed to generate emotion and energy.

Other expressions

Other aspects of musical expression are communicated by various body language signals. Staccato and legato can be differentiated by more or less 'spikey' movements. Phrasing is indicated by wide overhead arcs or by a smooth hand motion either forwards or side-to-side. A held note is often indicated by a hand held flat with palm up, and the end of a note is denoted by the closing of the palm, the pinching of finger and thumb, or by tracing a rapidly-twisted spiral with a finger or baton.

Rules of thumb

A good conductor aims to maintain eye contact with the ensemble as much as possible, encouraging eye contact in return and increasing the general dialogue between players/singers and conductor. Facial expressions are also important; all performers, but especially less experienced ones, respond well to encouraging expressions.

External links

- [http://www.theconcertband.com/conducting.htm Intro to Conducting]
- [http://thinkingapplied.com/conducting_folder/conducting1.htm What to Think About When You Conduct] (an orchestra) Category:Occupations in music ja:指揮

Unit circle

In mathematics, a unit circle is a circle with unit radius, i.e., a circle whose radius is 1. Frequently, especially in trigonometry, "the" unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Euclidean plane. The unit circle is often denoted S¹; the generalization to higher dimensions is the unit ball. If (x, y) is a point on the unit circle in the first quadrant, then x and y are the lengths of the legs of a right triangle whose hypotenuse has length 1. Thus, by the Pythagorean theorem, x and y satisfy the equation :

x^2 + y^2 = 1 \,\!

Since x² = (−x)² for all x, and since the reflection of any point on the unit circle about the x- or y-axis is also on the unit circle, the above equation holds for all points (x, y) on the unit circle, not just those in the first quadrant. One may also use other notions of "distance" to define other "unit circles"; see the article on normed vector space for examples.

Trigonometric functions on the unit circle

normed vector space The trigonometric functions cosine and sine may be defined on the unit circle as follows. If (x, y) is a point of the unit circle, and if the ray from the origin (0, 0) to (x, y) makes an angle t from the positive x-axis, (where the angle is measured in the counter-clockwise direction), then :

\cos(t) = x \,\!

\sin(t) = y \,\!

The equation x² + y² = 1 gives the relation :

\cos^2(t) + \sin^2(t) = 1 \,\!

The unit circle also gives an intuitive way of realizing that sine and cosine are periodic functions, with the identities :

\cos t = \cos(2\pi k+t) \,\!

\sin t = \sin(2\pi k+t) \,\!

:for any integer k. These identities come from the fact that the x- and y-coordinates of a point on the unit circle remain the same after the angle t is increased or decreased by any number of revolutions (1 revolution = 2π radians). integer When working with right triangles, sine, cosine, and other trigonometric functions only make sense for angle measures more than zero and less than π/2. However, using the unit circle, these functions have sensible, intuitive meanings for any real-valued angle measure. In fact, not only sine and cosine, but all of the six standard trigonometric functions — sine, cosine, tangent, cotangent, secant, and cosecant, as well as archaic functions like versine and exsecant — can be defined geometrically in terms of a unit circle, as shown at right.

Circle group

Complex numbers can be identified with points in the Euclidean plane, namely the number a + bi is identified with the point (a, b). Under this identification, the unit circle is a group under multiplication, called the circle group. This group has important applications in math and science; see circle group for more details.

Vector (spatial)

:This article discusses vectors that have a particular relation to the spatial coordinates. For a generalization, see vector space. In physics and in vector calculus, a spatial vector is a concept characterized by a magnitude, which is a scalar, and a direction (which can be defined in a 3-dimensional space by the Euler angles). Although it is often described by a number of "components", each of which is dependent upon the particular coordinate system being used, a vector is an object with properties which do not depend on the coordinate system used to describe it. A common example of a vector is force — it has a magnitude and an orientation in three dimensions (or however many spatial dimensions one has), and multiple forces sum according to the parallelogram law. A spatial vector can be formally defined by its relationship to the spatial coordinate system under rotations. Alternatively, it can be defined in a coordinate-free fashion via a tangent space of a three-dimensional manifold in the language of differential geometry. These definitions are discussed in more detail below. A spatial vector is a special case of a tensor and is also analogous to a four-vector in relativity (and is sometimes therefore called a three-vector in reference to the three spatial dimensions, although this term also has another meaning for p-vectors of differential geometry). Vectors are the building blocks of vector fields and vector calculus. The word vector is also now used for more general concepts (see also vector and generalizations below), but this article describes the original spatial meaning except where otherwise noted.

Definitions

Informally, a vector is a quantity characterized by a magnitude (in mathematics a number, in physics a number times a unit) and a direction, often represented graphically by an arrow. Examples are "moving north at 90 km/h" or "pulling towards the center of Earth with a force of 70 newtons". The notion of having a "magnitude" and "direction" is formalized by saying that the vector has components that transform like the coordinates under rotations. That is, if the coordinate system undergoes a rotation described by a rotation matrix R, so that a coordinate vector x is transformed to x' = Rx, then any other vector v is similarly transformed via v' = Rv. This ensures the invariance of the operations dot product, Euclidean norm, cross product, gradient, divergence, curl, and scalar triple product, and trivially for vector addition and subtraction, and scalar multiplication. The terms scalar and vector as used here include pseudoscalars and pseudovectors or axial vectors (see also below). Accordingly, let, for example, each of two vectors be expressed as three space coordinates, and apply the formula for the cross product, resulting in three coordinates, which represent a third vector. If we rewrite the two vectors in rotated coordinates, and apply the formula for the cross product again, then the result is the original cross product in terms of rotated coordinates. Also, let, for example, a vector field be expressed as three space coordinate functions of three variables, and apply the formula for the curl based on these functions, resulting in three additional functions, which represent a second vector field. If we rewrite the original vector field in terms of rotated position coordinates and correspondingly rotated coordinates for the vector function values, and apply the formula for the curl based on these functions, then the result is the rewritten version of the original curl: also in terms of rotated position coordinates and correspondingly rotated coordinates for the vector function values. The same applies for dot product, gradient, divergence, vector addition and scalar multiplication. For these, also reflection in a plane can be applied. The scalars involved should not be transformed (e.g. in the case of a rotation by 180°, the scalar should not be multiplied by -1). Thus even in 1D we have to distinguish scalars and vectors: 2 × 3 = 6 can be interpreted as a scalar multiplication or a dot product, but not as a product of two vectors. Similarly differentiation in 1D can be interpreted as a gradient or a divergence: one of the two functions is scalar and one a vector, and the argument is a vector, ensuring invariance under inversion of the vectors without changing the scalars. Since rotation of the three Cartesian coordinate axes changes the formulas the same as an inverse rotation of the field itself, we can also conclude:
- if the same rotation is applied to two vectors, then the cross product is correspondingly rotated, but the dot product remains the same
- rotation of a scalar field results in a correspondingly rotated vector field for the gradient
- rotation of a vector field results in a correspondingly rotated scalar field for the divergence and a correspondingly rotated vector field for the curl where rotation of a scalar field involves only rotation of the position vectors, while rotation of a vector field involves also a corresponding rotation of the vector field values. Note that the concept of corresponding rotations applies even if different coordinate systems are used for field values and position vectors, so that e.g. for one we multiply by an orthogonal matrix and for the other we add an angle to an angle coordinate. In order to use the usual formulas, e.g. to compute mechanical work, the x-axis of forces should be in the same direction as the x-axis of position, etc. When, as described above, coordinate rotations of position are accompanied by corresponding coordinate rotations of forces, this property is preserved. On the other hand, the origin of forces is simply at the zero force (no force), while the origin of position can be chosen as desired. For example, work depends on displacement, which is the difference of positions and therefore does not depend on the origin. Position and function value of a vector field are often, but not necessarily, expressed in similar coordinate systems. For example gravitational field strength due to a particular point mass may be

g_r = -9.8 (r/r_0)^ m/s^2

, with both the function value and the position vector in spherical coordinates. For the position vector the origin is chosen here at the center of the point mass; for the field strength the origin is simply at "zero field strength" anyway. How the other two coordinates are chosen does not matter in this case, because the field does not depend on them, and the field has no components in their directions. More generally, a vector is a tensor of contravariant rank one. In differential geometry, the term vector usually refers to quantities that are closely related to tangent spaces of a differentiable manifold (assumed to be three-dimensional and equipped with a positive definite Riemannian metric). (A four-vector is a related concept when dealing with a 4 dimensional spacetime manifold in relativity.) Examples of vectors include displacement, velocity, electric field, momentum, force, and acceleration. Vectors can be contrasted with scalar quantities such as distance, speed, energy, time, temperature, charge, power, work, and mass, which have magnitude, but no direction (they are invariant under coordinate rotations). The magnitude of any vector is a scalar. A related concept is that of a pseudovector (or axial vector). This is a quantity that transforms like a vector under proper rotations, but gains an additional sign flip under improper rotations. Examples of pseudovectors include magnetic field, torque, and angular momentum. (This distinction between vectors and pseudovectors is often ignored, but it becomes important in studying symmetry properties.) To distinguish from pseudo/axial vectors, an ordinary vector is sometimes called a polar vector. See also parity (physics). Sometimes, one speaks informally of bound or fixed vectors, which are vectors additionally characterized by a "base point". Most often, this term is used for position vectors (relative to an origin point). More generally, however, the physical interpretation of a particular vector can be parameterized by any number of quantities.

Examples in one dimension

A force may be "15N to the right", with coordinate 15N if the basis vector is to the right, and −15N if the basis vector is to the left. The magnitude of the vector is 15N in both cases. A displacement may be "4m to the right", with coordinate 4m if the basis vector is to the right, and −4m if the basis vector is to the left. The magnitude of the vector is 4m in both cases. The work done by the force in the case of this displacement is 60J in both cases. The force and displacement are vectors, the magnitudes are scalars, and the coordinates are neither.

Generalizations

In mathematics, a vector is any element of a vector space over some field. The spatial vectors of this article are a very special case of this general definition (they are not simply any element of R^d in d dimensions), which includes a variety of mathematical objects (algebras, the set of all functions from a given domain to a given linear range, and linear transformations). Note that under this definition, a tensor is a special vector!

Representation of a vector

Symbols standing for vectors are usually printed in boldface as a; this is also the convention adopted in this encyclopedia. Other conventions include

\vec

or a, especially in handwriting. Alternately, some use a tilde (~) placed under the vector. The length or magnitude or norm of the vector a is denoted by |a|. Vectors are usually shown in graphs or other diagrams as arrows, as illustrated below: Image:vecab.png Here the point A is called the tail, base, start, or origin; point B is called the head, tip, endpoint, or destination. The length of the arrow represents the vector's magnitude, while the direction in which the arrow points represents the vector's direction. If a vector is itself spatial, the length of the arrow depends on a dimensionless scale. If it represents e.g. a force, the "scale" is of physical dimension length/force. Thus there is typically consistency in scale among quantities of the same dimension, but otherwise scale ratios may vary; for example, if "1 newton" and "5 m" are both represented with an arrow of 2cm, the scales are 1:250 and 1m:50N respectively. Equal length of vectors of different dimension has no particular significance unless there is some proportionality constant inherent in the system that the diagram represents. Also length of a unit vector (of dimension length, not length/force, etc.) has no coordinate-system-invariant significance. In the figure above, the arrow can also be written as

\overrightarrow

or AB In order to calculate with vectors, the graphical representation is too cumbersome. Vectors in a n-dimensional Euclidean space can be represented as a linear combination of n mutually perpendicular unit vectors. In this article, we will consider R³ as an example. In R³, we usually denote the unit vectors parallel to the x-, y- and z-axes by i, j and k respectively. Any vector a in R³ can be written as a = a₁i + a₂j + a₃k with real numbers a₁, a₂ and a₃ which are uniquely determined by a. Sometimes a is then also written as a 3-by-1 or 1-by-3 matrix: :

= \begin
 a_1\\
 a_2\\
 a_3\\
\end

= \begin
 a_1 & a_2 & a_3 \\
\end

even though this notation suppresses the dependence of the coordinates a₁, a₂ and a₃ on the specific choice of coordinate system i, j and k.

Length of a vector

The length of the vector a = a₁i + a₂j + a₃k can be computed with the Euclidian norm :

\left\|\mathbf\right\|=\sqrt

which is a consequence of the Pythagorean theorem.

Vector equality

Two vectors are said to be equal if they have the same magnitude and direction. However if we are talking about bound vector, then two bound vectors are equal if they have the same base point and end point. For example, the vector i + 2j + 3k with base point (1,0,0) and the vector i+2j+3k with base point (0,1,0) are different bound vectors, but the same (unbounded) vector.

Vector addition and subtraction

Let a=a₁i + a₂j + a₃k and b=b₁i + b₂j + b₃k. The sum of a and b is: :

\mathbf+\mathbf
=(a_1+b_1)\mathbf
+(a_2+b_2)\mathbf
+(a_3+b_3)\mathbf

The addition may be represented graphically by placing the start of the arrow b at the tip of the arrow a, and then drawing an arrow from the start of a to the tip of b. The new arrow drawn represents the vector a + b, as illustrated below: Pythagorean theorem This addition method is sometimes called the parallelogram rule because a and b form the sides of a parallelogram and a + b is one of the diagonals. If a and b are bound vectors, then the addition is only defined if a and b have the same base point, which will then also be the base point of a + b. One can check geometrically that a + b = b + a and (a + b) + c = a + (b + c). The difference of a and b is: :

\mathbf-\mathbf
=(a_1-b_1)\mathbf
+(a_2-b_2)\mathbf
+(a_3-b_3)\mathbf

Subtraction of two vectors can be geometrically defined as follows: to subtract b from a, place the ends of a and b at the same point, and then draw an arrow from the tip of b to the tip of a. That arrow represents the vector a − b, as illustrated below: parallelogram If a and b are bound vectors, then the subtraction is only defined if they share the same base point which will then also become the base point of their difference. This operation deserves the name "subtraction" because (a − b) + b = a. In physics, vectors of different physical dimension may occur in the same diagram. However, adding or subtracting them (graphically or otherwise) is meaningless.

Scalar multiplication

A vector may also be multiplied by a real number r. In mathematics numbers are often called scalars to distinguish them from vectors, and this operation is therefore called scalar multiplication. The resulting vector is: :

r\mathbf=(ra_1)\mathbf
+(ra_2)\mathbf
+(ra_3)\mathbf

The length of ra is |r||a|. If the scalar is negative, it also changes the direction of the vector by 180^o. Two examples (r = -1 and r = 2) are given below: real number Here it is important to check that the scalar multiplication is compatible with vector addition in the following sense: r(a + b) = ra + rb for all vectors a and b and all scalars r. One can also show that a - b = a + (-1)b. The set of all geometrical vectors, together with the operations of vector addition and scalar multiplication, satisfies all the axioms of a vector space. Similarly, the set of all bound vectors with a common base point forms a vector space. This is where the term "vector space" originated. In physics, scalars also have a unit. The scale of acceleration in the diagram is e.g. 2 m/s² : cm, and that of force 5 N : cm. Thus a scale ratio of 2.5 kg : 1 is used for mass. Similarly, if displacement has a scale of 1:1000 and velocity of 0.2 cm : 1 m/s, or equivalently, 2 ms : 1, a scale ratio of 0.5 : s is used for time.

Unit vector

Main article: Unit vector A unit vector is any vector with a length of one. If you have a vector of arbitrary length, you can use it to create a unit vector. This is known as normalizing a vector. Unit vector To normalize a vector a = [a₁, a₂, a₃], scale the vector by the inverse of its length ||a||. That is: :

\mathbf=\frac=\frac\mathbf+\frac\mathbf+\frac\mathbf

Dot product

Main article: Dot product The dot product of two vectors a and b (sometimes called inner product, or, since its result is a scalar, the scalar product) is denoted by a·b and is defined as: :

\mathbf\cdot\mathbf
=\left\|\mathbf\right\|\left\|\mathbf\right\|\cos(\theta)

where ||a|| and ||b|| denote the norm (or length) of a and b, and θ is the measure of the angle between a and b (see trigonometric function for an explanation of cosine). Geometrically, this means that a and b are drawn with a common start point and then the length of a is multiplied with the length of that component of b that points in the same direction as a. This operation is often useful in physics; for instance, work is the dot product of force and displacement.

Cross product

The cross product (also vector product or outer product) differs from the dot product primarily in that the result of a cross product of two vectors is a vector. While everything that was said above can be generalized in a straightforward manner to more than three dimensions, the cross product is only meaningful in three dimensions (although a related product exists in seven dimensions - see below). The cross product, denoted a×b, is a vector perpendicular to both a and b and is defined as: :

\mathbf\times\mathbf
=\left\|\mathbf\right\|\left\|\mathbf\right\|\sin(\theta)\,\mathbf

where θ is the measure of the angle between a and b, and n is a unit vector perpendicular to both a and b. The problem with this definition is that there are two unit vectors perpendicular to both b and a. Which vector is the correct one depends upon the orientation of the vector space, i.e. on the handedness of the coordinate system. The coordinate system i, j, k is called right handed, if the three vectors are situated like the thumb, index finger and middle finger (pointing straight up from your palm) of your right hand. Graphically the cross product can be represented by this figure Image:crossproduct.png In such a system, a×b is defined so that a, b and a×b also becomes a right handed system. If i, j, k is left-handed, then a, b and a×b is defined to be left-handed. Because the cross product depends on the choice of coordinate systems, its result is referred to as a pseudovector. Fortunately, in nature cross products tend to come in pairs, so that the "handedness" of the coordinate system is undone by a second cross product. The length of a×b can be interpreted as the area of the parallelogram having a and b as sides.

Scalar triple product

The scalar triple product (also called the box product or mixed triple product) isn't really a new operator, but a way of applying the other two multiplication operators to three vectors. The scalar triple product is denoted by (a b c) and defined as: :

(\mathbf\ \mathbf\ \mathbf)
=\mathbf\cdot(\mathbf\times\mathbf)

It has three primary uses. First, the absolute value of the box product is the volume of the parallelepiped which has edges that are defined by the three vectors. Second, the scalar triple product is zero if and only if the three vectors are linearly dependent, which can be easily proved by considering that in order for the three vectors to not make a volume, they must all lie in the same plane. Third, the box product is positive if and only if the three vectors a, b and c are oriented like the coordinate system i, j and k. In coordinates, if the three vectors are thought of as rows, the scalar triple product is simply the determinant of the 3-by-3 matrix having the three vectors as rows. The scalar triple product is linear in all three entries and anti-symmetric in the following sense: : Technically, the scalar triple product is not a scalar, it is a pseudoscalar: under a coordinate inversion (x goes to −x), it flips sign.

Vectors as directional derivatives

A vector may also be defined as a directional derivative: consider a function

f(x^\alpha)

and a curve

x^\alpha (\sigma)

. Then the directional derivative of

f

is a scalar defined as

\frac = \frac\frac.

where the index

\alpha

is summed over the appropriate number of dimensions (e.g. from 1 to 3 in 3-dimensional Euclidian space, from 0 to 3 in 4-dimensional spacetime, etc.). Then consider a vector tangent to

x^\alpha (\sigma)

t^\alpha = \frac.

We can rewrite the directional derivative in differential form (without a given function

f

) as

\frac = t^\alpha\frac.

Therefore any directional derivative can be identified with a corresponding vector, and any vector can be identified with a corresponding directional derivative. We can therefore define a vector precisely: :

\mathbf \equiv a^\alpha \frac.

External links

- [http://wwwppd.nrl.navy.mil/nrlformulary/vector_identities.pdf Online vector identities] (PDF)
- Vectors at Wikibooks Category:Abstract algebra Category:Vector calculus Category:Linear algebra Category:Introductory physics ko:벡터 ja:ベクトル (数学)

Navigation

: This article concerns navigation in the sense of determination of position and direction on the surface of the Earth. See navigation (disambiguation) for other meanings. There are several traditions of navigation. In the pre-modern history of human migration and discovery of new lands by navigating the oceans, a few peoples have excelled as sea-faring explorers. Prominent examples are the Phoenicians, the Ancient Greeks, the Malays, the Persians, Arabians, the Norse and, perhaps more than any others, the peoples of the Pacific Ocean, particularly Polynesians and Micronesians.

Polynesian navigation

The Polynesian navigators routinely crossed thousands of miles of open ocean, to tiny inhabited islands, using only their own senses and knowledge, passed by oral tradition, from navigator to apprentice. In Eastern Polynesia, navigators, in order to locate directions at various times of day and year, memorized extensive facts concerning:
- the motion of specific stars, and where they would rise and set on the horizon of the ocean
- weather
- times of travel
- wildlife species (which congregate at particular positions)
- directions of swells on the ocean, and how the crew would feel their motion
- colors of the sea and sky, especially how clouds would cluster at the locations of some islands
- angles for approaching harbors These, and outrigger canoe construction methods, were kept as guild secrets. Generally each island maintained a guild of navigators who had very high status, since in times of famine or difficulty, only they could trade for aid or evacuate people. The guild secrets might have been lost, had not one of the last living navigators trained a professional small boat captain so that he could write a book. The first settlers of the Hawaiian Islands were said to have used these navigation methods to sail to the Hawaiian Islands from the Marquesas Islands. In 1973, the Polynesian Voyaging Society was established in Hawaii to research Polynesian navigation methods. They built a replica of an ancient double-hulled canoe called the Hokule'a, whose crew, in 1976, successfully navigated the Pacific Ocean from Hawaii to Tahiti using no instruments.
- [http://pvs.kcc.hawaii.edu/navigate/navigate.html Wayfinding Summary]
- [http://pvs.kcc.hawaii.edu/L2wayfind.html Wayfinding Main Page]

Western navigation

Modern methods

There are several different branches of navigation, including but not limited to:
- celestial navigation - navigation by observation of the sun, moon and stars
- pilotage - using visible natural and man made features such as sea marks and beacons
- dead reckoning - using compass and log to monitor expected progress on a journey
- waypoint navigation - using electronic equipment such as radio navigation and satellite navigation system to follow a course to a waypoint
- position fixing - determining current position by visual and electronic means
- collision avoidance using radar Knowing the ship's current position is the main problem for all navigators. Early navigators used pilotage, relying on local knowledge of land marks and coastal features, forcing all ships to stay close to shore. The magnetic compass allowing a course to be maintained and estimates of the ship's location to be calculated. Nautical charts were developed to record new navigational and pilotage information for use by other navigators. The development of accurate systems for taking lines of position based on the measurement of stars and planets with the sextant allowed ships to navigate the open ocean without needing to see land marks. Later developments included the placing of lighthouses and buoys close to shore to act a marine signposts identifying ambiguous features, highlighting hazards and pointing to safe channels for ships approaching some part of a coast after a long sea voyage. The invention of the radio lead to radio beacons and radio direction finders providing accurate land-based fixes even hundreds of miles from shore. These were made obsolete by satellite navigation systems. Traditional maritime navigation with a compass uses multiple redundant sources of position information to locate the ship's position. A navigator uses the ship's last known position and dead reckoning, based on the ship's logged compass course and speed, to calculate the current position. If the set and drift, due to tide and wind, can be determined, an estimated position can also be calculated. Periodically, the navigator needs to confirm the accuracy of the dead reckoning or estimated position calculations using position fixing techniques. This is done by correctly identifying reference points and measuring their bearings from the ship. These lines of position can be plotted on a nautical chart, with the intersection being the ship's current location. Addition lines of position can be measured in order to validate the results taken against other reference points. This is known as a fix. Celestial navigation systems are based on observation of the positions of the Sun, Moon and stars relative to the observer and a known location. Anciently the home port was used as the known location, currently the Greenwich Meridian or Prime Meridian is used as the known location for celestial charts. Navigators could determine their latitude by measuring the angular altitude of Polaris any time that it was visible (excepting, of course, in those southern latitudes from where it cannot be observed). Determining latitude by the sun was a little more difficult since the sun's altitude at noon during the year changes for a given location. Calculating the anticipated altitude of the sun for a given day and known position is done easily using Calculus. However, prior to the development and formulation of its key principles in the latter part of the 17th century by Isaac Newton and Gottfried Leibniz, tables of the sun's altitude during the year for a known port were used. The sun's angle over the horizon at noon was measured, and compared to the known angle at the same date as the known port. Local noon is easily determined by recording periodic readings of the altitude of the sun. Since periodic readings of the altitude will plot a sine wave, the maximum reading is the one used for local noon. Longitude is calculated as a time difference between the same celestial event at different locations. Noon was an easy event to observe. Local noon is determined while shooting the azimuth as described above. The time of the maximum altitude is easily determined by interpolating between periodic readings. The time of noon at the known location is carried by the navigator on an accurate clock. Then the local time of local noon is observed by the navigator. The difference of longitude is determined knowing that the sun moves to the west at 15 degrees per hour. The need for accurate navigation led to the development of progressively more accurate clocks. Once accurate clocks were available, detailed tables for celestial bodies were created so that navigational activities could take place anytime during the day or night, rather than at noon. In modern celestial navigation, a nautical almanac and trigonometric sight-reduction tables permit navigators to measure the Sun, Moon, visible planets or any of 57 navigational stars at any time of day or night. From a single sight, a time within a second and an estimated position, a position can be determined within a third of a mile (500 m). Conceptually, the angle to the celestial object establishes a ring of possible positions on the surface of the Earth. A second sighting on a different object establishes an intersecting ring. Usually the navigator knows his position well enough to pick which of the two intersections is the current position. The math required for sight reduction is simple addition and subtraction, if sight-reduction tables are available. The numerous celestial objects permit navigators to shoot through holes in clouds. Most navigation is performed with the sun and moon. Accurately knowing the time of an observation is important. Time is measured with a chronometer, a quartz watch or a shortwave radio broadcast from an atomic clock. A quartz wristwatch normally keeps time within a half-second per day. If it is worn constantly, keeping it near body heat, its rate of drift can be measured with the radio, and by compensating for this drift, a navigator can keep time to better than a second per month. Traditionally, three chronometers are kept in gimbals in a dry room near the center of the ship, and used to set a watch for the actual sight, so that the chronometers themselves do not risk exposure to the elements. Winding the chronometers was a crucial duty of the navigator. The angle is measured with a special optical instrument called a "sextant." Sextants use two mirrors to cancel the relative motion of the sextant. During a sight, the user's view of the star and horizon remains steady as the boat rocks. An arm moves a split image of the star relative to the split image of the horizon. When the image of the star touches the horizon, the angle can be read from the sextant's scale. Some sextants create an artificial horizon by reflecting a bubble. Inexpensive plastic sextants are available, though they have less accuracy than the more expensive metal models. The LORAN system is based on measuring the phase shift of radio waves sent simultaneously from a master and slave station. Signals from these two point establish a hyperbolic curve for possible positions. A third source along with dead-reckoning will generally resolve to a single position. GPS uses 3D trilateration based on measuring the time-of-flight of radio waves using the well-known speed of light to measure distance from at least three satelites. This can be accomplished using low-cost quartz clocks because the satellites send time correction signals to the GPS receivers.

History

In the West, navigation was at first performed exclusively by dead-reckoning, the process of estimating one's present position based on the navigators' experience with wind, tide and currents. Most sailors have always been able find absolute north from the stars, which currently rotate around Polaris, or by using a dual sundial called a diptych. When combined with a plumb bob, some diptychs could also determine latitude. Basically, when the diptych's two sundials indicated the same time, the diptych was aligned to the current latitude and true north. diptych Another early invention was the compass rose, a cross or painted panel of wood oriented with the pole star or diptych. This was placed in front of the helmsman. Latitude was determined with a "cross staff" an instrument vaguely similar to a carpenter's angle with graduated marks on it. Most sailors could use this instrument to take sun sights, but master navigators knew that sightings of Polaris were far more accurate, because they were not subject to time-keeping errors involved in finding noon. Time-keeping was by precision hourglasses, filled and tested to 1/4 of an hour, turned by the helmsman, or a young boy brought for that purpose. The most important instrument was a navigators' diary, later called a rutter. These were often crucial trade secrets, because they enabled travel to lucrative ports. The above instruments were a powerful technology, and appear to have been the technique used by ancient Cretan bronze-age trading empire. Using these techniques, masters successfully sailed from the eastern Mediterranean to the south coast of the British Isles. Some time later, around 300, the magnetic compass was invented in China. This let masters continue sailing a course when the weather limited visibility of the sky. China Around 400, metallurgy allowed construction of astrolabes graduated in degrees, which replaced the wooden latitude instruments for night use. Diptychs remained in use during the day, until shadowing astrolabes were constructed. After Isaac Newton published the Principia, navigation was transformed. Starting in 1670, the entire world was measured using essentially modern latitude instruments and the best available clocks. In 1730 the sextant was invented and navigators rapidly replaced their astrolabes. A sextant uses mirrors to measure the altitude of celestial objects with regard to the horizon. Thus, its "pointer" is as long as the horizon is far away. This eliminates the "cosine" error of an astrolabe's short pointer. Modern sextants measure to 0.2 minutes of arc, an error that translates to a distance of about 0.2 nautical miles (400 m). At first, the best available "clocks" were the moons of Jupiter, and the calculated transits of selected stars by the moon. These methods were too complex to be used by any but skilled astronomers, but they sufficed to map most of the world. A number of scientific journals during this period were started especially to chronicle geography. Later, mechanical chronometers enabled navigation at sea and in the air using relatively unskilled procedures. In the late 19th century Nikola Tesla invented radio and direction-finding was quickly adapted to navigation. Up until 1960 it was commonplace for ships and aircraft to use radio direction-finding on commercial stations in order to locate islands and cities within the last several miles of error. Around 1960, LORAN was developed. This used time-of-flight of radio waves from antennas at known locations. It revolutionized navigation by permitting semiautomated equipment to locate geographic positions to less than a half mile (800 m). An analogous system for aircraft, VOR and DME, was developed around the same time. At about the same, TRANSIT, the first satellite-based navigation system was developed. It was the first electronic navigation system to provide global coverage. Other radionavigation systems include:
- Decca
- Omega, a longwave system developed by the United States Navy
- Alpha, a longwave system developed by the Soviet Union In 1974, the first GPS satellite was launched. The GPS system now permits accurate geographic location with an error of only a few metres, and precision timing to less than a microsecond. GLONASS is a positioning system launched by the Soviet Union. It relies on a slightly different geodesic model of the Earth. Galileo is a competing system, that will be placed into service by the European Union.

"Point" measure of direction

A "point" is defined as one eighth of a right angle, and therefore equals exactly 11.25 degrees. For example, a bearing of northwest by north differs by one point from a northwest bearing, and by a point from a north-northwest one.

External links

- [http://www.wildernessmanuals.com/manual_6/chpt_2/index.html Navigation] - U.S Army Manual.
- [http://www.irbs.com/bowditch Bowditch Online] - complete online edition of Nathaniel Bowditch's American Practical Navigator Category:Navigation zh-min-nan:Tō-hâng ja:航海

Compass

:This article is about the navigational instrument. For other meanings, see compass (disambiguation). compass (disambiguation) A compass (or mariner's compass) is a navigational instrument for finding directions on the earth. It consists of a magnetised pointer free to align itself accurately with Earth's magnetic field. A compass provides a known reference direction which is of great assistance in navigation. The cardinal points are north, south, east and west. A compass can be used in conjunction with a clock and a sextant to provide a very accurate navigation capability. This device greatly improved maritime trade by making travel safer and more efficient. A compass can be any magnetic device using a needle to indicate the direction of the magnetic north of a planet's magnetosphere. Any instrument with a magnetized bar or needle turning freely upon a pivot and pointing in a northerly and southerly direction can be considered a compass. A compass dial is a small pocket compass with a sundial. A variation compass is a specific instrument of a delicate type of construction. It is used by observing variations of the needle. A gyrocompass or astrocompass can also be used to ascertain True north.

History of the navigational compass

Compasses were initially used in feng shui in ancient China. The first known use of Earth's magnetic field in this way occurred in ancient China as a spectacle. Arrows were cast similarly to dice. These magnetised arrows aligned themselves pointing north, impressing the audience. The earliest record of use of magnetic lodestone as a direction point was in a 4th century Chinese book: "Book of the Devil Valley Master" "Dream Pool Essays" written by Song Dynasty scholar Shen Kua in 1086 AD contained a detailed description of how geomancer magnetized a needle by rubbing its tip with lodestone, and hanged the magnetic needle with one single strain of silk with a bit of wax attached to the center of the needle. Shen Kua pointed out that the needle prepared this way some times pointed south, some times pointed north. The earliest record about the use of compass in navigation was Zhu Yu's book "Pingzhou Ke Tan" (Pingzhou Table Talks) of 1117 AD. :The navigator knows the geography, he watches the stars at night, watches the sun at day; when it is dark and cloudy, he watches the compass A pilot's compass handbook titled Shun Feng Xiang Song (Fair Winds for Escort) in the Oxford Bodleian Library contains great details about the use of compass in navigation. Bodleian Library Knowledge of the compass moved overland through the Arab countries and then to Europe sometime later in the 12th century. The compass, for example, is described and reported being used in the book Lubab ul-Albab for purposes of sea navigation around 1220. About 1358, there is a story about an English monk, Nicholas of Lynne, who served as a navigator due to his competence and knowledge in the "magnetic compass". (See Inventio Fortunata.) Prior to the introduction of the compass, wayfinding at sea was primarily done via celestial navigation, supplemented in some places by the use of soundings. Difficulties arose where the sea was too deep for soundings and conditions were continually overcast or foggy. Thus the compass was not of the same utility everywhere. For example, the Arabs could generally rely on clear skies in navigating the Persian Gulf and the Indian Ocean (as well as the predictable nature of the monsoons). This may explain in part their relatively late adoption of the compass. Mariners in the relatively shallow Baltic made extensive use of soundings. In the Mediterranean, however, the practice from ancient times had been to curtail sea travel between October and April, due in part to the lack of dependable clear skies during the Mediterranean winter (and much of the sea is too deep for soundings). With improvements in dead reckoning methods, and the development of better charts, this changed during the second half of the 13th century. By around 1290 the sailing season could start in late January or February, and end in December. The additional few months were of considerable economic importance; it enabled Venetian convoys, for instance, to make two round trips a year to the eastern Mediterranean, instead of one. Around the same time traffic between the Mediterranean and northern Europe increased, and one factor may be that the compass made traversal of the Bay of Biscay safer and easier.

Construction of a simple compass

A magnetic rod is required. This can be created by aligning an iron or steel rod with Earth's magnetic field and then tempering or striking it. However, this method produces only a weak magnet so other methods are preferred. This magnetised rod (or magnetic needle) is then placed on a low friction surface to allow it to freely pivot to align itself with the magnetic field. It is then labeled so the user can distinguish the north-pointing from the south-pointing end; in modern convention the north end is typically marked in some way, often by being painted red. Flavio Gioja (fl. 1302), an Italian marine pilot, is sometimes credited with perfecting the sailor's compass by suspending its needle over a fleur-de-lis design, which pointed north. He also enclosed the needle in a little box with a glass cover.

Modern navigational compasses

north Modern navigational compasses hold a magnetized needle inside a fluid-filled capsule; the fluid causes the needle to stop quickly rather than oscillate back and forth around magnetic north. Other features common on modern handheld compasses are a baseplate with rulings for measuring distances on maps, a rotating bezel for measuring bearings of distant objects, and a sighting mirror that lets the user see both the compass needle and a distant object at the same time. Mariner's compasses can have two or more magnetic needles permanently attached to a compass card. These move freely on a pivot. A lubber line, which can be a marking on the compass bowl or a small fixed needle indicates the ships heading on the compass card. Traditionally the card is divided into thirty-two points (known as rhumbs), although modern compasses are marked in degrees rather than cardinal points. The glass-covered box (or bowl) contains a suspended gimbal within a binnacle. This preserves the horizontal position. Large ships typically rely on a gyrocompass rather than a magnetic compass for navigation, and increasingly electronic fluxgate compasses are used on smaller vessels.

Solid state compasses

Small compasses found in clocks and other electronic gear are Solid-state electronics usually built out of two or three magnetic field sensors that provide data for a microprocessor. Using Trigonometry the correct heading relative to the compass is calculated.

Compass correction

:Main article magnetic deviation magnetic deviation A ship's compass must be corrected for errors, called compass deviation, caused by iron and steel in its structure and equipment. The ship is swung, that is rotated about a fixed point while its heading is noted by alignment with fixed points on the shore. A compass deviation card is prepared so that the navigator can convert between compass and magnetic headings. The compass can be corrected in three ways. First the lubber line can be adjusted so that it is aligned with the direction in which the ship travels, then the effects of permanent magnets can be corrected for by small magnets fitted within the case of the compass. The effect of non-ferromagnetic materials can be corrected by two iron balls mounted on either side of the compass binacle. The graph of the compass deviation can be understood using Fourier series. The coefficient

a_0

representing the error in the lubber line, while

a_1,b_1

the ferromagnetic effects and

a_2,b_2

the non-ferromagnetic component. Fluxgate compasses can be calibrated automatically, and can also be programmed with the correct local compass variation so as to indicate the true heading.

Points of the compass

Main article: Boxing the compass The mariner's compass card is divided into thirty-two equally spaced points. Four of these - east, west, north, and south - are the cardinal points, and the names of the others are derived from these. See also: Azimuth, Beam compass, coordinates, fluxgate compass, gyrocompass, Gyrosin compass, gyrostatic compass, inertial navigation system, radio compass, radio direction finder, pelorus

External links, resources, and references

- [http://geomag.usgs.gov USGS Geomagnetism Program]
- Amir Aczel, The Riddle of the Compass: The Invention that Changed the World, ISBN 0156007533
- Joseph Needham, Colin A. Ronan: The Shorter Science & Civilisation in China Vol 3 Chapter 1 Magnetism and Electricity.
- Science Friday, "[http://www.sciencefriday.com/pages/2002/May/hour2_053102.html The Riddle of the Compass]" (interview with Amir Aczel, first broadcast on NPR on May 31, 2002).
- Paul J. Gans, [http://scholar.chem.nyu.edu/tekpages/compass.html The Medieval Technology Pages: Compass]
- Frederic Lane, "The Economic Meaning of the Invention of the Compass", American Historial Review, vol. 68, pp. 605-617 (1963)
- www.howstuffworks.com "[http://science.howstuffworks.com/compass.htm How compasses work]" Category:Navigational equipment Category:Hiking equipment Category:Orientation Category:Measuring instruments ko:나침반 ja:方位磁針 simple:Compass th:เข็มทิศ

Left and right

, illustrating the x (right-left), y (forward-backward) and z (up-down) axes relative to a human being. In this illustration, right, up and forward are positive vectors, while left, down and backward are negative vectors. Left and right are defined from the boxer's perspective, rather than the viewer's.]] Left, right, forward, and back are the relative directions. They change as a person moves. People are often confused by the relative nature of these directions. "Left" for one person is different from "left" for another. Unlike the cardinal directions (North, South, East, and West), left and right are based solely on the viewer's perspective. The terms are derived from the fact that some 90% of the population are right-handed, that is, favour their right hand for a variety of tasks. The direction right is a cognate of "right" meaning correct or good. The word "left" comes from the Old English lyft, meaning weak. This dichotomy can also be seen in the words "dextrous" and "sinister", from the Latin words for right and left. In the vast majority of the population, the heart is slightly to the left of the body's centre line and the liver is to the right. Less visibly, the two hemispheres of the brain differ in function. Mirrors are said to reverse left and right. Strictly, this is not true: the image of one's left hand is on his or her left-hand side. However, it is seen at the position corresponding to that of the right hand of a real person opposite to him or her. Thus one can say that the mirror has the combined effect of turning you around and reversing left and right. One can also say that it reverses front and back side. If one takes a cube and writes "left", "right", "top", "bottom", "front", and "back" and holds it so that all sides face the correct direction, then hold a mirror behind it, without moving the cube or himself or herself, one will notice that "front" and "back" are swapped in the mirror image, but the other sides are still correctly labelled. In nautical usage, the halves of a ship are designated port (left) and starboard (right), relative to a person looking forward (towards the bow of the ship). In politics, "left" and "right" are labels applied to parts of a theoretical spectrum of opinion:
- Left is associated with liberalism, and, in the United States, with the Democratic Party. In the extreme, "left" can be associated with socialism and communism.
- Right is associated with conservativism, and, in the United States, with the Republican Party. In the extreme, "right" can be associated with fascism. This usage derives from the seating of parliamentary partisans during the French Revolution; see left-wing politics and right-wing politics. Category:Orientation

Example

zh:左和右 In this diagram, showing a road with right-hand traffic, the red car is to the left of the blue car. The blue car is, therefore, on the right-hand side. Should the blue car move backward, it would reach the position of the yellow car. For the red car to be where the green car is, it would have to move forward. The lane containing the green and red cars is the left lane, the lane with the yellow and blue cars is the right lane.

Direction (Alexander Technique

Direction is a term for a principal tool used in practicing the Alexander Technique. The principle underlying "Direction" is the substitution response, practiced to prevent habitual solutions from becoming more exaggerated and ingrained over time. Someone "directs" by visualizing or stating the more constructive means and manner of reaching their temporarily suspended objectives. "Direction" is used with a temporary suspension of goals (termed "inhibition"), because suspending goals postpones habitual remedies, allowing the newer "directions" to have a chance to happen. Some ways of "directing" are:
- to give consent for exploration and for "feeling odd";
- to learn to connect one's anatomical knowledge of the structural capacity of design for movement to the demonstration of it. This is sometimes done by stating the incremental steps of what you intend to do, as you begin the action, continuing to state them to yourself as you do them, as long as your attention can continue. Rather than gunning the motor by trying harder, and muscling your way through an activity, people who "direct" use strategic forethought to conduct and coordinate more constructive responses as they discover. "Directing" in the "Alexander Technique" means "mentally guiding the flow of force through the body", instead of "trying to do it". Since the body responds to "giving directions" by preparing to move at a miniscule response level, the body's reflexive coordination seems to handle the action by itself, gracefully and effortlessly. Continuing to "direct", builds easier responses over time, eventually completely replacing short-sighted, habitual knee-jerk solutions.

Direction - Social Democracy

The Direction - Social Democracy (Slovak: Smer - sociálna demokracia) party - before January 1 2005 called Direction (the Third Way) (Smer (tretia cesta)) - , often just Smer, is a political party in Slovakia, led by Robert Fico. It is considered to be left from the middle. It arose by splitting from the Party of the Democratic Left (Slovakia) (Strana demokratickej ľavice, SDĽ) in 1999 (Robert Fico being the most popular SDL member at that time) and quickly became one the most popular parties in Slovakia, while the popularity of the SDĽ, which was the successor of the original (i.e. pre-1990) Communist Party of Slovakia and was a government party from 1998 to 2002, was steadily decreasing. As of 2004, it was the third largest party in the National Council of the Slovak Republic, with 25 out of 150 seats. As of early 2005, it was in the first place among all Slovak political parties in opinion polls with 30% support. As from January 1, Smer merged with the small social democratic parties:
- Party of the Democratic Left (Slovakia) (see above)
- Social Democratic Alternative (Sociálnodemokratická alternatíva; a small modern-style social democratic party that split from the SDĽ somewhat later than Smer did), and
- Social Democratic Party of Slovakia (Sociálnodemokratická strana Slovenska; founded in 1990, the party became known under the leadership of Alexander Dubček). The resulting entity was renamed Smer - Social Democracy.

Slovakia

Slovakia (Slovak: Slovensko) is a landlocked republic in Central Europe. It borders the Czech Republic in the northwest, Poland in the north, Ukraine in the east, Hungary in the south, and Austria in the southwest.

Name

Slovakia is officially also called the Slovak Republic (in Slovak: Slovenská republika). The short form is as linguistically and historically as correct as the long one, just like France vs. French Republic, Slovenia vs. Republic of Slovenia etc. The recent practice, especially in economic texts, of using the name Slovak Republic instead of Slovakia, when the terms Hungary, Slovenia etc. are used in the same text, is therefore awkward, arising in analogy to the use of the term Czech Republic, but that is (partly) another problem (see Czech Republic, Czech lands).

History

Main article: History of Slovakia The original Slavic population settled the general territory of Slovakia in the 5th century. Slovakia was the centre of Samo's empire in the 7th century. A proto-Slovak state known as the Principality of Nitra arose in the 8th century and became part of the core of Great Moravian Empire (called Great Slovak Empire by a minority of Slovak authors) in 833. The high point of this (Proto-)Slovak empire came with the arrival of Cyril and Methodius and the expansion under King Svätopluk. After the disintegration of Great Moravian Empire in the early 10th century around the time of the Battle of Bratislava in 907, Slovakia became a part of the Kingdom of Hungary in the 11th-14th centuries. In the 10th century, the ethnic Slovak territory included the northern half of present-day Hungary, and in the 14th century it still extended to present-day northern central and northern eastern Hungary (down to present-day Vác (in Slovak Vacov), Visegrád (Višegrad/Vyšehrad), Miskolc (Miškovec)). A major share of the nobility in the kingdom was of Slovak origin. After the Ottoman Empire started its expansion into present-day Hungary in the early 16th century, the center of the Kingdom of Hungary (renamed Royal Hungary now) shifted towards Slovakia, and Bratislava (known as Pressburg/Pressporek/Posonium/Posony at that time) became its capital in 1536. By the end of the 18th century Slovakia's influence decreased. In the revolution of 1848-49 the Slovaks joined the Austrians to separate from the Kingdom of Hungary within the Austrian monarchy, but eventually they failed to achieve this aim. During the time of Austria-Hungary, i.e. 1867 - 1918, the Slovaks experienced one of the worst oppressions in their history in the form of Hungarisation (Magyarisation) promoted by the government. In 1918, Slovakia joined with the regions of Bohemia and neighbouring Moravia to form Czechoslovakia. During the chaos following the breakup of Austria-Hungary, a Slovak Soviet Republic was created for a very short time. Czechoslovakia lasted until it was broken up by the Munich Agreement of 1938. Slovakia became a separate republic that would be tightly controlled by Nazi Germany. After World War II, Czechoslovakia was reassembled and came under the influence of the Soviet Union and its Warsaw Pact from 1945 onward. The end of communist rule in Czechoslovakia in 1989 during the peaceful Velvet Revolution was followed once again by the country's dissolution, this time into two successor states. Slovakia and the Czech Republic went their separate ways after January 1, 1993. (Velvet Divorce) Slovakia became a member of the European Union in May 2004. See also: Bratislava - History, and History of Bratislava

Politics

Main article: Politics of Slovakia Politics of Slovakia Slovakia joined NATO on March 29, 2004 and the EU on May 1, 2004. There were Presidential elections in Slovakia on April 3, 2004 and April 17, 2004. The Slovak head of state is the president, elected by direct popular vote for a five-year term. Most executive power lies with the head of government, the prime minister, who is usually the leader of the major party or a majority coalition in parliament and appointed by the president. The remainder of the cabinet is appointed by the president on the recommendation of the prime minister. Slovakia's highest legislative body is the 150-seat unicameral National Council of the Slovak Republic (Národná rada Slovenskej republiky). Delegates are elected for four-year terms on the basis of proportional representation. Slovakia highest judicial body is the Constitutional Court (Ústavný súd), which rules on constitutional issues. The 13 members of this court are appointed by the president from a slate of candidates nominated by parliament. As a member of the United Nations since 1993, Slovakia was, on October 10, 2005, for the first time elected to a two-year term on the UN Security Council for 2006-2007. See also: List of rulers of Slovakia

Administrative divisions

Main article: Regions of Slovakia As for administrative division, Slovakia is subdivided into 8 kraje (singular - kraj, usually translated as regions, but actually meaning rather county), each of which is named after their principal city. As for territorial division and the definition of self-governing entities, since 2002, Slovakia is divided into eight Upper-Tier Territorial Units (sg. vyšší územný celok, pl. vyššie územné celky, abbr. VÚC) called samosprávny kraj (Self-governing (or: autonomous) Region): kraj # Bratislava Region (Bratislavský kraj) (see also Bratislava) # Trnava Region (Trnavský kraj) (see also Trnava) # Trenčín Region (Trenčiansky kraj) (see also Trenčín) # Nitra Region (Nitriansky kraj) (see also Nitra) # Žilina Region (Žilinský kraj) (see also Žilina) # Banská Bystrica Region (Banskobystrický kraj) (see also Banská Bystrica) # Prešov Region (Prešovský kraj) (see also Prešov) # Košice Region (Košický kraj) (see also Košice) (the word kraj can be replaced by samosprávny kraj in each case) The "kraje" are subdivided into many okresy (sg. okres, usually translated as districts). Slovakia currently has 79 districts. See also:
- List of traditional regions of Slovakia
- List of tourism regions of Slovakia List of tourism regions of Slovakia

Geography

Main article: Geography of Slovakia The Slovak landscape is noted primarily for its mountainous nature, with the Carpathian Mountains extending across most of the northern half of the country. Amongst them are the high peaks of the Tatra mountains, which are a popular skiing destination and home to many scenic lakes and valleys as well as the highest point in Slovakia, the Gerlachovský štít at 2,655 m. Lowlands are found in the southwestern (along the Danube) and southeastern parts of Slovakia. Major Slovak rivers, besides the Danube, are the Váh and the Hron. The Slovak climate is temperate, with relatively cool summers and cold, cloudy and humid winters.temperate

Economy

Main article: Economy of Slovakia Slovakia has mastered much of the difficult transition from a centrally planned economy to a modern market economy. The Slovak government made progress in 2001 in macroeconomic stabilisation and structural reform. Major privatisations are nearly complete, the banking sector is almost completely in foreign hands, and foreign investment has picked up. Slovakia's economy exceeded expectations in the early 2000s, despite recession in key export markets. Revival of domestic demand in 2002, partly due to a rise in real wages, offset slowing export growth to help drive the economy to its strongest expansion since 1998. Solid domestic demand boosted economic growth to 4.4 percent in 2002. Strong export growth, in turn, pushed economic growth to a still-strong 4.2 percent in 2003, despite a downturn in household consumption. Unemployment, rising from 14.9 percent at the end of 1998 to 19.2 percent at the end of 2001 (seasonally adjusted harmonised rate) during the radical reforms introduced by the Slovak government since 1999, decreased again to 12.7 percent (March 2005). Inflation dropped from an average annual rate of 12.0 percent in 2000 to just 3.3 percent in the election year 2002, but it rose again in 2003-2004 due to necessary increases in taxes and regulated prices. Nonetheless, CPI fell below 3 percent in 2005. Slovakia would like to adopt the Euro currency on 1 January 2009, although the public sector deficit needs to be cut in the draft budget from its current 3.8 percent of GDP to below 3 percent in order for this to be possible. In a survey of the German Chamber of Commerce held in March 2004, as much as 50 percent of German enterpreneurs chose Slovakia as the best place for investment.

Demographics

Main article: Demographics of Slovakia Demographics of Slovakia The majority of the inhabitants of Slovakia are ethnically Slovak (86 percent). Hungarians are the largest ethnic minority (10 percent) and are concentrated in the southern and eastern regions of the country. Other ethnic groups include Roma, Czechs, Ruthenians, Ukrainians and Germans. The percentage of Roma is 1.7% according to the last census (that is based on the own definition of the Roma), but around 5.6% based on interviews with municipality representatives and mayors (that is based on the definition of the remaining population). Note however that in the case of the 5.6%, the above percentages of Hungarians and Slovaks are lower by 4 percentage points in sum. The Slovak constitution guarantees freedom of religion. The majority of Slovak citizens (68.9 percent) practice Roman Catholicism; the second-largest group are people without confession (12.96 percent). About 6.93 percent belong to Lutheranism and 4.1 percent are Greek Catholic,Calvinism has 2.0 percent, other and non-registered churches 1.1 percent i.e., Eastern Catholic and some 0.9 percent are Eastern Orthodox. About 2,300 Jews remain of the estimated pre-WWII population of 120,000. The official state language is Slovak, a member of the Slavic languages, but Hungarian is also widely spoken in the south and enjoys a co-official status in some regions.

Holidays

- National holidays in Slovakia
- Remembrance days in Slovakia

Culture

Main article: Culture of Slovakia
- Music of Slovakia
- Slovak literature

Miscellaneous topics

- Coat of Arms of Slovakia
- Communications in Slovakia
- Education in Slovakia
- Foreign relations of Slovakia
- History of ice hockey in Slovakia
- List of national parks of Slovakia
- List of rivers in Slovakia
- List of rulers of Slovakia
- List of Slovaks
- List of tourism regions of Slovakia
- List of towns in Slovakia
- List of traditional regions of Slovakia
- Military of Slovakia
- Ski and winter sports in Slovakia
- Slovak euro coins
- Slovenský skauting
- Transportation in Slovakia
- Namedays in the Slovak Republic

External links

- [http://www.government.gov.sk/english The Slovak Republic Government Office]
- [http://www.foreign.gov.sk/En/index.html Ministry of Foreign Affairs of the Slovak Republic]
- [http://www.settlement.org/cp/english/slovakia/ Cultural profile of Slo

Direction

Film director

See also

External links

Conducting

History of conducting

Conducting technique

Beat and tempo

Tempo

Dynamics

Entries

Other expressions

Rules of thumb

Further reading

See also

External links

Unit circle

Trigonometric functions on the unit circle

Circle group

See also

Vector (spatial)

Definitions

Examples in one dimension

Generalizations

Representation of a vector

Length of a vector

Vector equality

Vector addition and subtraction

Scalar multiplication

Unit vector

Dot product

Cross product

Scalar triple product

Vectors as directional derivatives

See also

External links

Navigation

Polynesian navigation

Western navigation

Modern methods

History

"Point" measure of direction

See also

External links

Compass

History of the navigational compass

Construction of a simple compass

Modern navigational compasses

Solid state compasses

Compass correction

Points of the compass

External links, resources, and references

Left and right

Example

Direction (Alexander Technique

Direction - Social Democracy

See also

Slovakia

Name

History

Politics

Administrative divisions

Geography

Economy

Demographics

Holidays

Culture

Miscellaneous topics

Further reading

External links