vector

vector, in physics, a quantity that has both magnitude and direction. It is typically represented by an arrow whose direction is the same as that of the quantity and whose length is proportional to the quantity’s magnitude. Although a vector has magnitude and direction, it does not have position. That is, as long as its length is not changed, a vector is not altered if it is displaced parallel to itself.

In contrast to vectors, ordinary quantities that have a magnitude but not a direction are called scalars. For example, displacement, velocity, and acceleration are vector quantities, while speed (the magnitude of velocity), time, and mass are scalars.

vector parallelogram for addition and subtractionOne method of adding and subtracting vectors is to place their tails together and then supply two more sides to form a parallelogram. The vector from their tails to the opposite corner of the parallelogram is equal to the sum of the original vectors. The vector between their heads (starting from the vector being subtracted) is equal to their difference.

To qualify as a vector, a quantity having magnitude and direction must also obey certain rules of combination. One of these is vector addition, written symbolically as A + B = C (vectors are conventionally written as boldface letters). Geometrically, the vector sum can be visualized by placing the tail of vector B at the head of vector A and drawing vector C—starting from the tail of A and ending at the head of B—so that it completes the triangle. If A, B, and C are vectors, it must be possible to perform the same operation and achieve the same result (C) in reverse order, B + A = C. Quantities such as displacement and velocity have this property (commutative law), but there are quantities (e.g., finite rotations in space) that do not and therefore are not vectors.

right-hand rule for vector cross product
right-hand rule for vector cross productThe ordinary, or dot, product of two vectors is simply a one-dimensional number, or scalar. In contrast, the cross product of two vectors results in another vector whose direction is orthogonal to both of the original vectors, as illustrated by the right-hand rule. The magnitude, or length, of the cross product vector is given by |v| |w| sin θ, where θ| is the angle between the original vectors v and w.

The other rules of vector manipulation are subtraction, multiplication by a scalar, scalar multiplication (also known as the dot product or inner product), vector multiplication (also known as the cross product), and differentiation. There is no operation that corresponds to dividing by a vector. See vector analysis for a description of all of these rules.

Although vectors are mathematically simple and extremely useful in discussing physics, they were not developed in their modern form until late in the 19th century, when Josiah Willard Gibbs and Oliver Heaviside (of the United States and England, respectively) each applied vector analysis in order to help express the new laws of electromagnetism, proposed by James Clerk Maxwell.

The Editors of Encyclopaedia BritannicaThis article was most recently revised and updated by Erik Gregersen.
Citation Information
Article Title: vector
Website Name: Encyclopaedia Britannica
Publisher: Encyclopaedia Britannica, Inc.
Date Published: 27 May 2020
URL: https://www.britannica.comhttps://www.britannica.com/science/vector-physics
Access Date: November 21, 2024