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Kilometre Per Second

Kilometre per second

Metre per second (U.S. spelling: meter per second) is an SI derived unit of both speed (scalar) and velocity (vector), defined by distance in metres divided by time in seconds. The symbol is m/s, or equivalently, m s^-1.

Conversions

1 metre per second is equivalent to:
- 3.2808 feet per second
- 2.2369 miles per hour
- 3.6 km/h

United States

:For alternative meanings, see the disambiguation page for US, USA, United States, or American. The United States of America is a federal democratic republic situated primarily in central North America. It comprises 50 states and one federal district, and has several territories. It is also referred to, with varying formality, as the United States, the U.S., the U.S.A., the States, or simply and most commonly, America. The official founding date of the United States is July 4, 1776, when the Second Continental Congress—representing thirteen British colonies—adopted the Declaration of Independence. However, the structure of the government was profoundly changed in 1788, when the states replaced the Articles of Confederation with the United States Constitution. The date on which each of the fifty states adopted the Constitution is typically regarded as the date that state "entered the Union" (became part of the United States). Since the mid-20th century, following World War II, the United States has emerged as a dominant global influence in economic, political, military, scientific, technological, and cultural affairs.

Geography and climate

The United States shares land borders with Canada (to the north) and Mexico (to the south), and territorial water boundaries with Canada, Russia, the Bahamas, and numerous smaller nations. It is otherwise bounded by the Pacific Ocean and the Bering Sea, in the west; the Arctic Ocean, in the northernmost areas; and the Atlantic Ocean, the Gulf of Mexico, and the Caribbean Sea, in the eastern and southeastern areas. Forty-eight of the states are in the single region between Canada and Mexico; this group is referred to, with varying precision and formality, as the continental or contiguous United States, sometimes abbreviated CONUS, and as the Lower 48. Alaska, which is not included in the term contiguous United States, is at the northwestern end of North America, separated from the Lower 48 by Canada. The archipelago of Hawaii is in the Pacific Ocean. The capital city, Washington, District of Columbia is a federal district located on land donated by the state of Maryland. (Virginia also donated land, but it was returned in 1847.) The United States also has overseas territories with varying levels of independence and organization. When inland water is included in the total area, only Russia and Canada are larger than the United States; if inland water is excluded, China ranks third and the U.S. ranks fourth. The United States' total area is 3,718,711 square miles (9,631,418 km²), of which land makes up 3,537,438 square miles (9,161,923 km²) and water makes up 181,273 square miles (469,495 km²). The United States' landscape is one of the most varied among those of the world's nations: among its many features are temperate forestland and rolling hills, on the east coast; mangrove, in Florida; the Great Plains, in the center of the country; the Mississippi–Missouri river system; the Great Lakes, four of the five of which are shared with Canada; the Rocky Mountains, west of the Great Plains; deserts and temperate coastal zones, west of the Rocky Mountains; and temperate rain forests, in the Pacific northwest. Alaska's tundra, and the volcanic, tropical islands of Hawaii add to the geographic diversity. Hawaii The climate varies along with the landscape, from tropical in Hawaii and southern Florida to tundra in Alaska and atop some of the highest mountains. Most of the North and East experience a temperate continental climate, with warm summers and cold winters. Most of the South experiences a subtropical humid climate with mild winters and long, hot, humid summers. Rainfall decreases markedly from the humid forests of the Eastern Great Plains to the semi-arid shortgrass prairies on the high plains abutting the Rocky Mountains. Arid deserts, including the Mojave, extend through the lowlands and valleys of the southwest, from westernmost Texas to California and northward throughout much of Nevada. Some parts of California have a Mediterranean climate. Rainforests line the windward mountains of the Pacific Northwest from Oregon to Alaska.

History

American history started with the migration of people from Asia across the Bering land bridge approximately 12,000 years ago following large animals that they hunted into the Americas. These Native Americans left evidence of their presence in petroglyphs, burial mounds, and other artifacts. It is estimated that 2-9 million people lived in the territory now occupied by the U.S. before European contact, and the subsequent introduction of foreign diseases such as small pox that greatly diminished the native populations. Some advanced societies were the Anasazi of the southwest, who inhabited Chaco Canyon, and the Woodland Indians, who built Cahokia, located near present-day St Louis, a city with a population of 40,000 at its peak in AD 1200. Vikings first visited North America around 1000, but did not settle permanently. Following the discovery voyages of Christopher Columbus around 1492, other Europeans began to explore and settle there. During the 1500s and 1600s, the Spanish settled parts of the present-day Southwest and Florida, founding St. Augustine, Florida in 1565 and Santa Fe (in what is now New Mexico) in 1607. The first successful English settlement was at Jamestown, Virginia, also in 1607. Within the next two decades, several Dutch settlements, including New Amsterdam (the predecessor to New York City), were established in what are now the states of New York and New Jersey. In 1637, Sweden established a colony at Fort Christina (in what is now Delaware), but lost the settlement to the Dutch in 1655. This was followed by extensive British settlement of the east coast. The British colonists remained relatively undisturbed by their home country until after the French and Indian War, when France ceded Canada and the Great Lakes region to Britain. Britain then imposed taxes on the 13 colonies, widely regarded by the colonists as unfair because they were denied representation in the British Parliament. Tensions between Britain and the colonists increased, and the thirteen colonies eventually rebelled against British rule. British Parliament, George Washington (1789-1797).]] In 1776, the 13 colonies split from Great Britain and formed the United States, the world's first constitutional and democratic federal republic, after their Declaration of Independence of that year, and the Revolutionary War (1775 to 1783). The original political structure was a confederation in 1777, ratified in 1781 as the Articles of Confederation. After long debate, this was supplanted by the Constitution in 1789, forming a more centralized federal government. Prior to all these was the Albany Congress in 1754, in which a union was first seriously proposed. From early colonial times, there was a shortage of labor, which encouraged unfree labor, particularly indentured servitude and slavery. In the mid-19th century, a major division occurred in the United States over the issue of states' rights and the expansion of slavery. The northern states had become opposed to slavery, while the southern states saw it as necessary for the continued success of southern agriculture and wanted it expanded to the territories. Several federal laws were passed in an attempt to settle the dispute, including the Missouri Compromise and the Compromise of 1850. The dispute reached a crisis in 1861, when seven southern states seceded¹ from the Union and formed the Confederate States of America, leading to the Civil War. Soon after the war began, four more southern states seceded. During the war, Abraham Lincoln issued the Emancipation Proclamation, mandating the freedom of all slaves in states in rebellion, though full emancipation did not take place until after the end of the war in 1865, the dissolution of the Confederacy, and the Thirteenth Amendment took effect. The Civil War effectively ended the question of a state's right to secede, and is widely accepted as a major turning point after which the federal government became more powerful than state governments. Thirteenth Amendment). The title of the painting, from a 1726 poem by Bishop Berkeley, was a phrase often quoted in the era of Manifest Destiny, expressing a widely held belief that civilization had steadily moved westward throughout history. [http://americanart.si.edu/t2go/1lw/1931.6.1.html (more)] ]] During the 19th century, many new states were added to the original 13 as the nation expanded across the continent. Manifest Destiny was a philosophy that encouraged westward expansion in the United States. As the population of the Eastern states grew and as a steady increase of immigrants entered the country, settlers moved steadily westward across North America. In the process, the U.S. displaced most American Indian nations. This displacement of American Indians continues to be a matter of contention in the U.S. with many tribes attempting to assert their original claims to various lands. In some areas American Indian populations were reduced by foreign diseases contracted through contact with European settlers, and US settlers acquired those emptied lands. In other instances American Indians were removed from their traditional lands by force. Though some would say the U.S. was not a colonial power until the Spanish-American War when it acquired Puerto Rico, Guam and the Philippines, the dominion exercised over land in North America the United States claimed is essentially colonial. The Philippines became independent in 1946. During this period, the nation also became an industrial power. This continued into the 20th century, which has been termed "the American Century" because of the nation's overriding influence on the world. The US became a center for innovation and technological development; major technologies that America either developed or was greatly involved in improving include the telephone, television, computer, the Internet, nuclear weapons, nuclear power, aviation, and aeronautics. In addition to the Civil War, another major traumatic experience for the nation was the Great Depression (1929 to 1939). The nation has also taken part in several major foreign wars, including World War I and World War II (in both of which the US later joined the Allies). During the Cold War, the US was a major player in the Korean War and Vietnam War, and, along with the Soviet Union, was considered one of the world's two "superpowers". With the collapse of the Soviet Union, the US emerged as the world's leading economic and military power. Beginning in the 1990s, the United States became very involved in police actions and peacekeeping, including actions in Kosovo, Haiti, Somalia and Liberia, and the first Persian Gulf War driving Iraq out of Kuwait. After attacks on the World Trade Center and the Pentagon on September 11, 2001, the United States and other allied nations found themselves involved in what has come to be called the "War on Terrorism," which has primarily encompassed military actions in both Afghanistan and Iraq.

Government

Iraq of the United States.]]

Republic and suffrage

The United States is an example of a constitutional republic, with a government composed of and operating through a set of limited powers imposed by its design and enumerated in the United States Constitution. Specifically, the nation operates as a presidential democracy. There are three levels of government: federal, state, and local. Officials of each of these levels are either elected by eligible voters via secret ballot or appointed by other elected officials. Americans enjoy almost universal suffrage from the age of 18 regardless of race, sex, or wealth. There are some limits, however: felons are disenfranchised and in some states former felons are likewise. Furthermore, the national representation of territories and the federal district of Washington, DC in Congress is limited: residents of the District of Columbia are subject to federal laws and federal taxes but their only Congressional representative is a non-voting delegate.

Federal government

The federal government is the national government, comprising the Legislative Branch (led by Congress), the Executive Branch (led by the President), and the Judicial Branch (led by the Supreme Court). These three branches were designed to apply checks and balances on each other. The Constitution limits the powers of the federal government to defense, foreign affairs, the issuing and management of currency, the management of trade and relations between the states, and the protection of human rights. In addition to these explicitly stated powers, the federal government—with the assistance of the Supreme Court—has gradually extended these powers into such areas as welfare and education, on the basis of the "necessary and proper" clause of the Constitution.

The Congress

necessary and proper The Congress of the United States is the legislative branch of the federal government of the United States. It is bicameral, comprising the House of Representatives and the Senate. The House of Representatives consists of 435 members, each of whom represents a congressional district and serves for a two-year term. House seats are apportioned among the states by population; in contrast, each state has two Senators, regardless of population. There are a total of 100 senators, who serve six-year terms. The powers of Congress are limited to those enumerated in the Constitution; all other powers are reserved to the states and the people. The Constitution also includes the necessary-and-proper clause, which grants Congress the power to "make all laws which shall be necessary and proper for carrying into execution the foregoing powers."

The President

necessary-and-proper clause At the top level of the executive branch is the President of the United States. The President and Vice-President are elected as 'running mates' for four-year terms by the Electoral College, for which each state, as well as the District of Columbia, is allocated a number of seats based on its representation (or ostensible representation, in the case of D. C.) in both houses of Congress (see U.S. Electoral College). The relationship between the President and the Congress reflects that between the English monarchy and parliament at the time of the framing of the United States Constitution. Congress can legislate to constrain the President's executive power, even with respect to his or her command of the armed forces; however, this power is used only very rarely—a notable example was the constraint placed on President Richard Nixon's strategy of bombing Cambodia during the Vietnam War. The President cannot directly propose legislation, and must rely on supporters in Congress to promote his or her legislative agenda. The President's signature is required to turn congressional bills into law; in this respect, the President has the power—only occasionally used—to veto congressional legislation. Congress can override a presidential veto with a two-thirds majority vote in both houses. The ultimate power of Congress over the President is that of impeachment or removal of the elected President through a House vote, a Senate trial, and a Senate vote. The threat of using this power has had major political ramifications in the cases of Presidents Andrew Johnson, Richard Nixon, and Bill Clinton. The President makes around 2,000 executive appointments, including members of the Cabinet and ambassadors, which must be approved by the Senate; the President can also issue executive orders and pardons, and has other Constitutional duties, among them the requirement to give a State of the Union address to Congress once a year. Although the President's constitutional role may appear to be constrained, in practice, the office carries enormous prestige that typically eclipses the power of Congress: the Presidency has justifiably been referred to as 'the most powerful office in the world'. The Vice President is first in the line of succession, and is the President of the Senate ex officio, with the ability to cast a tie-breaking vote. The members of the President's Cabinet are responsible for administering the various departments of state, including the Department of Defense, the Justice Department, and the State Department. These departments and department heads have considerable regulatory and political power, and it is they who are responsible for executing federal laws and regulations. George W. Bush is the 43rd President, currently serving his second term.

The Courts

George W. Bush The highest court is the Supreme Court, which consists of nine justices. The court deals with federal and constitutional matters, and can declare legislation made at any level of the government as unconstitutional, nullifying the law and creating precedent for future law and decisions. Below the Supreme Court are the courts of appeals, and below them in turn are the district courts, which are the general trial courts for federal law. Separate from, but not entirely independent of, this federal court system are the individual court systems of each state, each dealing with its own laws and having its own judicial rules and procedures. A case may be appealed from a state court to a federal court only if there is a federal question; the supreme court of each state is the final authority on the interpretation of that state's laws and constitution.

State and local governments

supreme court of each state. Note that Alaska and Hawaii are shown at different scales, and that the Aleutian Islands and the uninhabited Northwestern Hawaiian Islands are omitted from this map.]] The state governments have the greatest influence over people's daily lives. Each state has its own written constitution and has different laws. There are sometimes great differences in law and procedure between the different states, concerning issues such as property, crime, health, and education. The highest elected official of each state is the Governor. Each state also has an elected legislature (bicameral in every state except Nebraska), whose members represent the different parts of the state. Of note is the New Hampshire legislature, which is the third-largest legislative body in the English-speaking world, and has one representative for every 3,000 people. Each state maintains its own judiciary, with the lowest level typically being county courts, and culminating in each state supreme court, though sometimes named differently. In some states, supreme and lower court justices are elected by the people; in others, they are appointed, as they are in the federal system. The institutions that are responsible for local government are typically town, city, or county boards, making laws that affect their particular area. These laws concern issues such as traffic, the sale of alcohol, and keeping animals. The highest elected official of a town or city is usually the mayor. In New England, towns operate directly democratically, and in some states, such as Rhode Island and Connecticut, counties have little or no power, existing only as geographic distinctions. In other areas, county governments have more power, such as to collect taxes and maintain law enforcement agencies.

Political divisions

With the Declaration of Independence, the thirteen colonies proclaimed themselves to be nation states modeled after the European states of the time. Although considered as sovereigns initially, under the Articles of Confederation of 1781 they entered into a "Perpetual Union" and created a fully sovereign federal state, delegating certain powers to the national Congress, including the right to engage in diplomatic relations and to levy war, while each retaining their individual sovereignty, freedom and independence. But the national government proved too ineffective, so the administrative structure of the government was vastly reorganized with the United States Constitution of 1789. Under this new union, the continued status of the individual states as sovereign nation states fell into dispute in 1861, as several states attempted to secede from the union; in response, then-President Abraham Lincoln claimed that such secession was illegal, and the result was the American Civil War. Since the Union victory in 1865, the independent status of the individual states has not been broached again by any state, and the status of each state within the union has been deemed by mainstream officials and academics to be settled as being subordinate to the union as a whole. In subsequent years, the number of states grew steadily due to western expansion, the purchase of lands by the national government from other nation states, and the subdivision of existing states, resulting in the current total of 50. The states are generally divided into smaller administrative regions, including counties, cities and townships. The United States–Canadian border is the longest undefended political boundary in the world. The U.S. is divided into three distinct sections:
- the "continental United States," also known as "the Lower 48" and more accurately termed the conterminous, coterminous or contiguous United States
- Alaska, which is physically connected only to Canada
- the archipelago of Hawaii, in the central Pacific Ocean. The United States also holds several other territories, districts, and possessions, notably the federal district of the District of Columbia, which is the nation's capital, and several overseas insular areas, the most significant of which are American Samoa, Guam, the Northern Mariana Islands, Puerto Rico, and the United States Virgin Islands. The Palmyra Atoll is the United States' only incorporated territory; it is unorganized and uninhabited. The United States Navy has held a base at a portion of Guantanamo Bay, Cuba, since 1898. The United States government possesses a lease to this land, which only mutual agreement or United States abandonment of the area can terminate. The present Cuban government of Fidel Castro disputes this arrangement, claiming Cuba was not truly sovereign at the time of the signing. The United States argues this point moot because Cuba apparently ratified the lease post-revolution, and with full sovereignty, when it cashed one rent check in accordance with the disputed treaty.

Foreign relations and military

sovereign] The immense military and economic dominance of the United States has made foreign relations an especially important topic in its politics, with considerable concern about the image of the United States throughout the world. Reactions towards the United States by other nationalities are often strong, ranging from uninhibited admiration and mimicking of all things American to anti-Americanism. US foreign policy has swung about several times over the course of its history between the poles of strict isolationism and imperialism and everywhere in between. Three of the nation's four military branches are administered by the Department of Defense: the Army, the Navy (including the Marine Corps), and the Air Force. The Coast Guard falls under the jurisdiction of the Department of Homeland Security in peacetime, but is placed under the Department of the Navy in time of war. The combined United States armed forces consist of 1.4 million active duty personnel, along with several hundred thousand each in the Reserves and the National Guard. Military conscription ended in 1973. The United States Armed forces are considered to be the most powerful military (of any sort) on Earth and their force projection capabilities are unrivaled by any other nation. The 2005 defense budget amounted to $401.7 billion, which is an increase of 4% over 2004 and of 35% since 2001. Over 50% of that number is spent in research & development. (For comparison, in 2004 the European Union (considered as the second-largest military force) had a combined total of 1.6 million troops, and a defense budget of €160 billion, with less than 10% of that being spent on R&D.)

Largest cities

The United States has dozens of major cities, including 11 of the 55 global cities of all types — with three "alpha" global cities: New York City, Los Angeles, and Chicago. The figures expressed below are for populations within city limits. A different ranking is evident when considering U.S. metro area populations, although the top three would be unchanged. Note that some cities not listed (such as Atlanta, Boston, Las Vegas, Miami, Nashville, New Orleans, Seattle, and Washington, D.C.) are still considered important on the basis of other factors and issues, including culture, economics, heritage, and politics. The twenty largest cities, based on the United States Census Bureau's 2004 estimates, are as follows:

Economy

The United States has the largest single-country economy in the world, with a per-capita gross domestic product of $40,100. In this market-oriented economy, private individuals and business firms make most of the decisions, and the federal and state governments buy needed goods and services predominantly in the private marketplace. gross domestic product The largest industry of the U.S. is now service, which employs roughly three quarters of the U.S. work force. The United States has many natural resources, including oil and gas, metals, and such minerals as gold, soda ash, and zinc. In agriculture, the U.S. is a top producer of, among other crops, corn, soy beans, and wheat; the United States is a net exporter of food. The U.S. manufacturing sector produces goods such as, cars, airplanes, steel, and electronics, among many others. Economic activity varies greatly from one part of the country to another, with many industries being largely dependent on a certain city or region; New York City is the center of the American financial, publishing, broadcasting, and advertising industries; Silicon Valley is the country’s primary location for high-technology companies, while Los Angeles is the most important center for film production. The Midwest is known for its reliance on manufacturing and heavy industry, with Detroit, Michigan, serving as the center of the American automotive industry; the Great Plains are known as the "breadbasket" of America for their tremendous agricultural output; the intermountain region serves as a mining hub and natural gas resource; the Pacific Northwest for fish and timber, while Texas is largely associated with the oil industry; the Southeast is a major hub for both medical research and the textiles industry. Several countries continue to link their currency to the dollar or even use it as a currency (such as Ecuador), although this practice has subsided since the collapse of the Bretton Woods system. Many markets are also quoted in dollars, such as those of oil and gold. The dollar is also the predominant reserve currency in the world, and more than half of global reserves are in dollars. The largest trading partner of the United States is Canada (19%), followed by China (12%), Mexico (11%), and Japan (8%). More than 50% of total trade is with these four countries. In 2003, the United States was ranked as the third most visited tourist destination in the world; its 40,400,000 visitors ranked behind France's 75,000,000 and Spain's 52,500,000. Labor unions have existed since the 19th century, and grew large and powerful from the 1930s to the 1950s. See Labor history of the United States. Since 1970 they have shrunk in the private sector and now cover fewer than 8% of the workers. However union membership has grown rapidly in the public sector, especially among teachers, nurses, police, postal workers, and municipal clerks. There have been few strikes in recent years. The United States' imports exceed exports by 80%, leading to an annual trade deficit of $700,000,000,000, or 6% of gross domestic product. It is the largest debtor nation in the world, with total gross foreign debt of over $13,000,000,000,000 (2005 estimate); and it absorbs more than 50% of global savings annually. Since the 1980s, the U.S. has increased the use of neoliberal economic policies that reduce government intervention and reduce the size of the welfare state, backing away from the more interventionist Keynsian economic policies that had been in favor since the Great Depression. As a result, the United States provides fewer government-delivered social welfare services than most industrialized nations, choosing instead to keep its tax burden lower and relying more heavily on the free market and private charities. Sixteen states and the District of Columbia have minimum wages higher than the national level ($5.15 per-hour), including the highest, Washington State at $7.35. Twenty-six states are the same as the federal level; two--Ohio and Kansas--are below; and six do not have state laws. America's wealth is relatively highly concentrated. The average C.E.O. earns 500 times the typical amount a worker grosses, this is up from 25 times in the late 1970s. In terms of wealth the top 1% of Americans own 40% of all assets and 50.1% of the country's income goes to the top twenty percent of households. Average wages for the majority of employees have been largely stagnating since the 1970s. America's poverty line defined as a family of four earning less than $19,157 is at 12.7% of the general population. Approximately one out of every five children in the United States grows up below the official poverty line. Among racial groups; African Americans have the lowest median income while Asians had the highest. Regionally, the southern states had the lowest median incomes while the West Coast and New England had the highest. The current Federal Reserve Chairman Alan Greenspan remarked that the U.S.’s growing income inequality since the 1970s is, "not the type of thing which a democratic society - a capitalist democratic society - can really accept without addressing."[http://www.csmonitor.com/2005/0614/p01s03-usec.html?s=itm] However, Greenspan also noted, "...you can look at the system and say it's got a lot of problems to it, and sure it does. It always has. But you can't get around the fact that this is the most extraordinarily successful economy in history."

Transportation

Alan Greenspan ]] Because the United States is a relatively young nation, most of the development of U.S. cities has taken place since the invention of the automobile. To link its vast territory, the United States built a network of high-capacity, high-speed highways, of which the most important element is the Interstate Highway system, commissioned in the 1950s by President Dwight D. Eisenhower and modeled after the German Autobahn. The United States also has a transcontinental rail system, which is used for moving freight across the lower forty-eight states. Passenger rail service is provided by Amtrak, which serves forty-six of the lower forty-eight states. Many cities in the United States have extensive mass-transit systems. New York City operates one of the world's largest and most heavily used subway systems. The regional rail and bus networks that extend into Long Island, New Jersey, Upstate New York, and Connecticut are among the most heavily used in the world. Air travel is often preferred for destinations over 300 miles (500 kilometers) away. In terms of passengers, seventeen of the world's thirty busiest airports in 2004 were in the U.S., including the world's busiest, Hartsfield-Jackson Atlanta International Airport; in terms of cargo, in the same year, twelve of the world's thirty busiest airports were in the U.S., including the world's busiest, Memphis International Airport. There are several major seaports in the United States; the three busiest are the Port of Los Angeles, California; the Port of Long Beach, California; and the Port of New York and New Jersey. Others include Houston, Texas; Charleston, South Carolina; Savannah, Georgia; Miami, Florida; Portland, Oregon; San Francisco, California; Boston, Massachusetts; Philadelphia, Pennsylvania; and Seattle, Washington; plus, outside the contiguous forty-eight states, Anchorage, Alaska, and Honolulu, Hawaii.

Society

Demographics

Hawaii The mean center of the U.S. population continues to drift farther west and south. The fastest growing region is the western United States followed by the southern portion. According to Census 2000, the states that saw the greatest increases from 1990 were: Nevada (66.3%), Arizona (40%), Colorado (30.6%), Utah (29.6%), Idaho (28.5%), Georgia (26.4%), Florida (23.5%), Texas (22.8%), North Carolina (21.4%), and Washington (21.1%). [http://www.census.gov/population/cen2000/phc-t2/tab03.pdf]

Ethnicity and race

:Main article: Racial demographics of the United States The United States is a very racially diverse country. According to the 2000 census, it has 31 ethnic groups with at least one million members each, and numerous others represented in smaller amounts. The majority of Americans descend from white European immigrants who arrived at the establishment of the first colonies (most after Reconstruction). This majority--69.1% in 2000--decreases each year, and is expected to become a plurality within a few decades. The most frequently stated European ancestries are German (15.2%), Irish (10.8%), English (8.7%), Italian (5.6%) and Scandinavian (3.7%). Many immigrants also hail from Slavic countries such as Poland and Russia. Other significant immigrant populations came from eastern and southern Europe and French Canada. Russia Hispanics from Mexico and South and Central America are the largest minority group in the country, comprising 12.5% of the population (2000 census). People of Mexican descent made up 7.3% of the population in the 2000 census, and this proportion is expected to increase significantly in the coming decades. About 12.3% (2000 census) of the American people are African Americans (Blacks). African Americans are spread throughout the country, but their presence is largest in the South. Asian Americans--including Native Hawaiians and Pacific Islanders--are a third significant minority (3.7% of the population in 2000). Most Asian Americans are concentrated on the West Coast and Hawaii. The largest groups are immigrants or descendants of emigrants from the Philippines, China, India, Vietnam, South Korea, and Japan. Indigenous peoples in the United States, such as American Indians and Inuit, make up 0.9% of the population (2000 census). About 35% live on Indian reservations.

Religion

Polls estimate that just under 80 percent of Americans are Christians of various denominations. The other 20 percent comprises other religions such as Hinduism, Judaism, Islam, and Buddhism, other various faiths, and those without a specific religion. The United States is noteworthy among developed nations for its relatively high level of religiosity. According to a 2004 Gallup poll, about 44% of Americans attend a religious service at least once a week. However, this rate is not uniform across the country; attendance is more common in the Bible Belt—composed largely of Southern and Midwestern states—than in the Northeast and West Coast. In the Southern states, Baptists are the largest group, followed by Methodists; Roman Catholics are dominant in the Northeast and in large parts of the Midwest due to their being settled by descendants of Catholic immigrants from Europe (such as Germany, Ireland, Italy, and Poland) or other parts of North America (mainly Quebec and Puerto Rico). The rest of the country for the most part has a complex mixture of various Christian groups.

Education

West Coast's home at Monticello and the University of Virginia (library building shown above, and designed by Jefferson), the only collegiate campus on the list. Both sites are located in Charlottesville, Virginia.]] In the United States, education is a state, not federal, responsibility, and the laws and standards vary considerably. However, the federal government, through the Department of Education, is involved with funding of some programs and exerts some influence through its ability to control funding. In most states, all students must attend mandatory schooling starting with kindergarten, which children normally enter at age 5, and following through 12th grade, which is normally completed at age 18

SI derived unit

SI derived units are part of the SI system of measurement units and are derived from the seven SI base units.

Dimensionless derived units

The following SI units are actually dimensionless ratios, formed by dividing two identical SI units. They are therefore considered by the BIPM to be derived. Formally, their SI unit is simply the number 1, but they are given these special names, for use whenever the lack of a unit might be confusing.

Derived units with special names

Base units can be put together to derive units of measurement for other quantities. Some have been given names.

Other quantities and units

Conversion between kelvins and degrees Celsius

A change in temperature of 1°C is equal to a change in temperature of 1K. Temperature in °C = Temperature in kelvins - 273.15 Thus, one could think of the Kelvin scale as the same as the Celsius scale, with its zero point moved down to absolute zero. This perspecitive is historically accurate; however, it has become more convenient to fix the standard for the kelvin, and thus the Celsius scale is derived from that standard (i.e., it now depends on absolute zero and the triple point of water with a 0.01 K offset — the boiling point of water no longer has anything to do with the official definition of degrees Celsius). Temperature differences are often measured in degrees Celsius; however, it doesn't matter: differences in temperature are equivalent whether kelvins or degrees Celsius are used. Therefore, a change in temperature (ΔT), when expressed in an equation, can be calculated using either kelvins or degrees celsius so long as one is consistent.

References

- I. Mills, Tomislav Cvitas, Klaus Homann, Nikola Kallay, IUPAC: Quantities, Units and Symbols in Physical Chemistry, 2nd edition (June 1993), Blackwell Science Inc (p. 72) ko:SI 유도 단위 ja:SI組立単位 Category:SI units Category:SI derived units

Speed

:For alternate uses, see special education or speed (disambiguation). Speed (symbol: v) is the rate of motion, or equivalently the rate of change of position, expressed as distance d moved per unit of time t. Speed is a scalar quantity with dimensions distance/time; the equivalent vector quantity to speed is known as velocity. Speed is measured in the same physical units of measurement as velocity, but does not contain the element of direction that velocity has. Speed is thus the magnitude component of velocity. Units of speed include:
- metres per second, (symbol m/s), the SI derived unit
- kilometres per hour, (symbol km/h)
- miles per hour, (symbol mph)
- knots (nautical miles per hour, symbol kt)
- Mach, where Mach 1 is the speed of sound; Mach n is n times as fast. ::Mach 1 = ~343 m/s = ~1235 km/h = ~768 mi/h (see the speed of sound for more detail)
- speed of light in vacuum (symbol c) is one of the natural units ::c = 299,792,458 m/s
- [other important conversions] ::1 m/s = 3.6 km/h ::1 mph = 1.609 km/h ::1 knot = 1.852 km/h = 0.514 m/s Vehicles often have a speedometer to measure the speed. The rate of change of speed with respect to time is termed acceleration.

Average speed

Speed as a physical property represents primarily instantaneous speed. In real life we often use average speed (denoted

\tilde

), which is rate of total distance (or length) and time interval. For example, if you go 60 miles in 2 hours, your average speed during that time is 60/2 = 30 miles per hour, but your instantaneous speed may have varied. In mathematical notation: :

\tilde = \frac.

Instantaneous speed defined as a function of time on interval

[t_0, t_1]

gives average speed:
:

\tilde = \frac

while instant speed defined as a function of distance (or length) on interval

[l_0, l_1]

gives average speed:
:

\tilde = \frac

It is often intuitively expected that going half a distance with speed

v_

and second half with speed

v_

, produce total average speed

\tilde = \frac

. The correct value is

\tilde = \frac

(Note that the first is arithmetic mean while the second is harmonic mean).
Average speed can be derived also from speed distribution function (either in time or on distance): :

v \sim D_t\; \Rightarrow \; \tilde = \int v D_t(v) \, dv

v \sim D_l\; \Rightarrow \; \tilde = \frac

Cultural significance

Speed or swiftness of motion plays a significant role in human culture, see racing. It is complementary to grace, precision and strength, e.g. in dancing or martial arts. Animals symbolizing speed are the horse (PIE
- ek'vos is etymologically derived from
- ok'u- "swift"), birds, especially raptors such as the hawk, and cats, e.g. the lynx (see e.g. Flos Duellatorum). The swiftest land animal is the cheetah, reaching running speeds of up to 110km/h.

External links

- [http://calc.skyrocket.de/en/ Online Unit Converter - Conversion of many different units]
- Category:Physical quantity ko:속력 ja:速さ nb:Fart simple:Speed th:ความเร็ว

Scalar

Scalar is a concept that has meaning in mathematics, physics, and computing. A simple definition is that a scalar is a quantity which only specifies magnitude (i.e. a numerical value and unit), unlike a vector which has both magnitude and direction. For example, speed (180 km/h) is a scalar, while velocity (180 km/h north) is a vector. While this is a useful definition, it is not quite complete. A scalar is more completely defined as a magnitude which does not change under a change of coordinate system. In the above example, suppose the velocity vector has two components (e.g, 180 km/h north and 0 km/h east). Each component has a magnitude, yet they are not scalars, because they change when the coordinate system used to calculate them changes. (e.g to 180/√2 km/h northwest and 180/√2 km/h northeast.) Similar considerations hold for the mathematical definition. It is only for the computer definition that "scalar" simply means a single number. The word scalar derives from the English word "scale" for a range of numbers, which in turn is derived from scala (Latin for "ladder"). According to a citation in the Oxford English Dictionary the first usage of the term (by W. R. Hamilton in 1846) described it as: :"The algebraically real part may receive, according to the question in which it occurs, all values contained on the one scale of progression of numbers from negative to positive infinity; we shall call it therefore the scalar part." Hamilton's usage actually describes his quaternion-based notation, which (in modern terms) represented rotations by a scalar, the real part of the quaternion, and vectors by the other three parts. Quaternions are widely used in spacecraft attitude determination and control, because they are not subject to the singularities of Euler angles and have only four components, while a rotation matrix has nine components to represent only three angles.

In physics

In physics, a scalar is a physical quantity which assumes a single value which is independent of the coordinate system being used to describe the physical system. In this sense it is a "real" quantity and not an artifact of the coordinate system. For example, the distance between two points in space is a scalar. It does not depend on one's choice of coordinate system. A physical quantity is expressed as the product of a numerical value and a physical unit, not just a number. It does not depend on the unit distance (1 km is the same as 1000 m), although the number depends on the unit. Thus distance does not depend on the length of the base vectors of the coordinate system. Also, other changes of the coordinate system may affect the formula for computing the scalar (for example, the Euclidean formula for distance in terms of coordinates relies on the basis being orthonormal), but not the scalar itself. In this sense, physical distance deviates from the definition of metric in not being just a real number; however it satisfies all other properties. The same applies for other physical quantities which are not dimensionless. A scalar field is a scalar-valued function of position, again independent of the coordinate system. A vector is a physical entity which has a magnitude which is a scalar, but in addition, in contrast with a scalar, has a direction. The components of a vector as such are not scalars, since they change with a change of coordinate system; a scalar field may however for one choice of the coordinate system be equal to a particular component. Examples of scalar quantities:
- electric charge and charge density (the latter nonrelativistically; in relativity it must be combined with current density to comprise a 4-vector)
- relativistic distance
- mass and mass density (the latter nonrelativistically; in relativity it must be made part of the energy tensor in combination with momentum density and pressure)
- speed, but not velocity or momentum
- temperature
- energy and energy density (the latter nonrelativistically) A related concept is a pseudoscalar, which is invariant under proper rotations but (like a pseudovector) flips sign under improper rotations. One example is the scalar triple product (see vector), and thus the signed volume. Another example is magnetic charge (as it is mathematically defined, regardless of whether it exists physically).

In mathematics

In mathematics, the meaning of scalar depends on the context; it can refer to real numbers or complex numbers or rational numbers, or to members of some other specified field. Generally, when a vector space over the field F is studied, then F is called the field of scalars and members of F are called scalars. More generally, a scalar for a module over a ring, is simply an element of the ring. This happens in manifold theory, where the tangent bundle forms a module over the algebra of real functions on the manifold. Since spacetime is supposed to be a manifold, the physical and mathematical concepts agree. A scalar is a tensor of rank zero.

In computing

In computing scalar refers to variables that can hold only one value at a time, as distinct from arrays, list or other containers which are variables that can hold many values at the same time.

Velocity

This article is about velocity in physics. For other meanings, see velocity (disambiguation). The velocity of an object is simply its speed in a particular direction. Note that both speed and direction are required to define a velocity.

Explanation

The velocity (v) is an physical quantity of the motion. A change in an object's velocity can therefore arise from either a change in its speed or in its direction. For example an aeroplane that is circling at a constant speed of 200km/h is changing its velocity because it is continously changing its direction. A aeroplane that is taking-off may go from zero to 200km/h in a straight line and so would also be changing its velocity. A change in velocity is called an acceleration. Objects are only accelerated if a force is applied to them. (The amount of acceleration depends the size of the force and the mass of the object being shifted, see Newton's Second Law of Motion.) In the case of the circling aeroplane, the pilot banks to use the force of lift from the wings to change direction. In another example the Space Shuttle orbits the earth at a constant speed but is constantly changing its velocity because of the circular orbit. In this case the force causing the acceleration is provided by the earth's gravity acting on the shuttle. The average speed v of an object moving a distance d during a time interval t is described by the formula: :

v = \frac

Acceleration is the rate of change of an object's velocity over time. The average acceleration of a of an object whose speed changes from v_i to v_f during a time interval t is given by: :

a = \frac

Where

v_i

= an object's initial velocity and

v_f

= the object's final velocity over a period of time t

Formal description

Velocity (symbol: v) is a vector measurement of the rate and direction of motion. The scalar absolute value (magnitude) of velocity is speed. Velocity can also be defined as rate of change of displacement or just as the rate of displacement, depending on how the term displacement is used. It is thus a vector quantity with dimension length/time. In the SI (metric) system it is measured in metre per second The instantaneous velocity vector v of an object that has position at time t is given by x(t) can be computed as the derivative :

v= = \lim_

The instantaneous acceleration vector a of an object that has position at time t is given by x(t) is :

\mathbf = \frac   = \frac

The equation for an object's velocity can be obtained mathematically by taking the integral of the equation for its acceleration beginning from some initial period time $t_0$ to some point in time later $t_n$ . The final velocity v_f of an object which starts with velocity v_i and then accelerates at constant acceleration a for a period of time t is: :

v_f = v_i + a t

The average velocity of an object undergoing constant acceleration is (v_i + v_f)/2. To find the displacement d of such an accelerating object during a time interval t, substitute this expression into the first formula to get: :

d = t \times \frac

When only the object's initial velocity is known, the expression :

d = v_i t + \frac

can be used. These basic equations for final velocity and displacement can be combined to form an equation that is independent of time, also known as Torricelli's Equation: :

v_f^2 = v_i^2 + 2 a d

The above equations are valid for both classical mechanics and special relativity. Where classical mechanics and special relativity differ is in how different observers would describe the same situation. In particular, in classical mechanics, all observers agree on the value of t and the transformation rules for position create a situation in which all non-accelerating observers would describe the acceleration of an object with the same values. Neither is true for special relativity. The kinetic energy (energy of motion) of a moving object is linear with both its mass and the square of its velocity: :

E_ = \begin \frac \end mv^2

The kinetic energy is a scalar quantity.

Polar coordinates

In polar coordinates, a two-dimensional velocity can be decomposed into a radial velocity, defined as the component of velocity away from or toward the origin, and transverse velocity, the component of velocity along a circle centred at the origin, and equal to the distance to the origin times the angular velocity. Angular momentum in scalar form is the distance to the origin times the transverse speed, or equivalently, the distance squared times the angular speed, with a plus or minus to distinguish clockwise and anti-clockwise direction. If forces are in the radial direction only, as in the case of a gravitational orbit, angular momentum is constant, and transverse speed is inversely proportional to the distance, angular speed is inversely proportional to the distance squared, and the rate at which area is swept out is constant. These relations are known as Kepler's laws of planetary motion.

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)

Metre

:This article is about the unit of length. For other uses of metre or meter, see meter (disambiguation). The metre (Commonwealth English) or meter (American English) (symbol: m) is the SI base unit of length. It is defined as the length of the path travelled by light in absolute vacuum during a time interval of 1/299,792,458 of a second. Adding SI prefixes to metre creates multiples and submultiples; for example kilometre (1000 metres; kilo- = 1000) and millimetre (one thousandth of a metre; milli- = 1 / 1 000).

Conversions

1 metre is equivalent to:
- exactly 1/0.9144 yards (approximately 1.0936 yards)
- exactly 1/0.3048 feet (approximately 3.2808 feet)
- exactly 10000/254 inches (approximately 39.370 inches)

History

The word metre is from the Greek metron (μετρον), "a measure" via the French mètre. Its first recorded usage in English is from 1797. In the 18th century, there were two favoured approaches to the definition of the standard unit of length. One suggested defining the metre as the length of a pendulum with a half-period of one second. The other suggested defining the metre as one ten-millionth of the length of the earth's meridian along a quadrant (one-fourth the polar circumference of the earth). In 1791, the French Academy of Sciences selected the meridional definition over the pendular definition because of the slight variation of the force of gravity over the surface of the earth, which affects the period of a pendulum. In 1793, France adopted the metre, with this definition, as its official unit of length. Although it was later determined that the first prototype metre bar was short by a fifth of a millimetre due to miscalculation of the flattening of the earth, this length became the standard. So, the circumference of the Earth through the poles is approximately forty million metres. Earth in a vacuum.]] In the 1870s and in light of modern precision, a series of international conferences were held to devise new metric standards. The Metre Convention (Convention du Mètre) of 1875 mandated the establishment of a permanent International Bureau of Weights and Measures (BIPM: Bureau International des Poids et Mesures) to be located in Sèvres, France. This new organisation would preserve the new prototype metre and kilogram when constructed, distribute national metric prototypes, and would maintain comparisons between them and non-metric measurement standards. This organisation created a new prototype bar in 1889 at the first General Conference on Weights and Measures