vector - Encyclopædia Britannica

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vector

Encyclopdia Britannica

vector, in mathematics, a quantity [1] that has both magnitude [2] and direction [3] but not

position. Examples of such quantities are velocity [4] and acceleration [5]. In their modern

form, vectors appeared late in the 19th century when Josiah Willard Gibbs [6] and Oliver

Heaviside [7] (of the United States [8] and Britain, respectively) independently developed

vector analysis[9]

to express the new laws[10]

of electromagnetism[11]

discovered by

the Scottish physicist James Clerk Maxwell [12]. Since that time, vectors have become

essential in physics [13], mechanics [14], electrical engineering [15], and other sciences to

describe forces mathematically.

Vectors may be visualized as directed line [16] segments whose lengths are their

magnitudes. Since only the magnitude and direction of a vector matter, any directed

segment may be replaced by one of the same length and direction but beginning at

another point[17]

, such as the origin of a coordinate system[18]

. Vectors are usuallyindicated by a boldface letter, such as v. A vectors magnitude, or length, is indicated by

|v|, or v, which represents a one-dimensional quantity (such as an ordinary number)

known as a scalar [19]. Multiplying a vector by a scalar [20] changes the vectors length

but not its direction, except that multiplying by a negative number [21] will reverse the

direction of the vectors arrow. For example, multiplying a vector by 1/2 will result in a

vector half as long in the same direction, while multiplying a vector by 2 will result in a

vector twice as long but pointed in the opposite direction.

Two vectors can be added [22] or subtracted [23]. For example, to add or subtract vectors

v and w graphically (see the diagram), move each to the origin and complete the

parallelogram formed by the two vectors; v + w is then one diagonal vector of the

parallelogram, and v w is the other diagonal vector.

http://www.britannica.com/EBchecked/topic/624306/v

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There are two different ways of multiplying [24] two vectors together. The cross, or

vector, product results in another vector that is denoted by v w. The cross product [25]

magnitude is given by |v w| = vw sin , where is the smaller angle [26] between the

vectors (with their tails placed together). The direction of v w is perpendicular to

both v and w, and its direction can be visualized with the right-hand rule [27], as shown in

the figure. The cross product is frequently used to obtain a normal (a line

perpendicular) to a surface[28]

at some point, and it occurs in the calculation[29]

of

torque [30] and the magnetic force [31] on a moving charged particle [32].

The other way of multiplying two vectors together is called a dot product [33], or

sometimes a scalar product[34]

because it results in a scalar. The dot product is given by

v

w =vw

cos

, where

is the smaller angle between thevector

s. The dot product isused to find the angle between two vectors. (Note that the dot product is zero when the

vectors are perpendicular.) A typical physical application is to find the work W

performed by a constant [35] force Facting on a moving object d; the work is given by

W= Fdcos .

http://www.britannica.com/EBchecked/topic/486142/quantity1.

http://www.britannica.com/EBchecked/topic/357547/magnitude2.

http://www.britannica.com/EBchecked/topic/164982/direction3.

http://www.britannica.com/EBchecked/topic/624901/velocity4.

http://www.britannica.com/EBchecked/topic/2810/acceleration5.

http://www.britannica.com/EBchecked/topic/233207/J-Willard-Gibbs6.

http://www.britannica.com/EBchecked/topic/258889/Oliver-Heaviside7.

http://www.britannica.com/EBchecked/topic/616563/United-States8.

http://www.britannica.com/EBchecked/topic/624327/vector-analysis9.

http://www.britannica.com/EBchecked/topic/411752/new-law10.

http://www.britannica.com/EBchecked/topic/183324/electromagnetism11.

http://www.britannica.com/EBchecked/topic/370621/James-Clerk-Maxwell12.

http://www.britannica.com/EBchecked/topic/458757/physics13.


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http://www.britannica.com/EBchecked/topic/371907/mechanics14.

http://www.britannica.com/EBchecked/topic/182783/electrical-and-electronics-engineering15.

http://www.britannica.com/EBchecked/topic/341961/line16.

http://www.britannica.com/EBchecked/topic/466303/point17.

http://www.britannica.com/EBchecked/topic/136400/coordinate-system18.

http://www.britannica.com/EBchecked/topic/526284/scalar19.

http://www.britannica.com/EBchecked/topic/526284/scalar20.

http://www.britannica.com/EBchecked/topic/408039/negative-number21.

http://www.britannica.com/EBchecked/topic/5457/addition22.

http://www.britannica.com/EBchecked/topic/571137/subtraction23.

http://www.britannica.com/EBchecked/topic/397199/multiplication24.

http://www.britannica.com/EBchecked/topic/624362/vector-product25.

http://www.britannica.com/EBchecked/topic/24777/angle26.

http://www.britannica.com/EBchecked/topic/503479/right-hand-rule27.

http://www.britannica.com/EBchecked/topic/575004/surface28.

http://www.britannica.com/EBchecked/topic/89152/calculation29.

http://www.britannica.com/EBchecked/topic/600049/torque30.

http://www.britannica.com/EBchecked/topic/357171/magnetic-force31.

http://www.britannica.com/EBchecked/topic/106443/charged-particle32.

http://www.britannica.com/EBchecked/topic/288565/inner-product33.

http://www.britannica.com/EBchecked/topic/288565/inner-product34.

http://www.britannica.com/EBchecked/topic/133764/constant35.


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vector - Encyclopædia Britannica

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