quantity having both magnitude and direction; it may be represented by a directed line segment. Many physical quantities are vectors, e.g., force, velocity, and momentum. Thus, in specifying a force, one must state not only how large it is but also in what direction it acts.

Representation and Reference Systems

The simplest representation of a vector is as an arrow connecting two points. Thus, AB designates the vector represented by an arrow from point A to point B, while BA designates a vector of equal magnitude in the opposite direction, from B to A. In order to compare vectors and to operate on them mathematically, however, it is necessary to have some reference system that determines scale and direction. Cartesian coordinates are often used for this purpose. In the plane, two axes and unit lengths along each axis serve to determine magnitude and direction throughout the plane. For example, if the point A mentioned above has coordinates (2,3) and the point B coordinates (5,7), the size and position of the vector are thus determined. The size of the vector in the x-direction is found by projecting the vector onto the x-axis, i.e., by dropping perpendicular line segments to the x-axis. The length of this projection is simply the difference between the x-coordinates of the two points A and B, or 5−2=3. This is called the x-component of the vector. Similarly, the y-component of the vector is found to be 7−3=4. A vector is frequently expressed by giving its components with respect to the coordinate axes; thus, our vector becomes [3,4].

Knowledge of the components of a vector enables one to compute its magnitude—in this case, 5, from the Pythagorean theorem [(3²+4²)^1/2=5)]—and its direction from trigonometry, once the lengths of the sides of the right triangle formed by the vector and its components are known. (Trigonometry can also be used to find the component of the vector as projected in some direction other than the x-axis or y-axis.) Since the vector points from A to B, both its components are positive; if it pointed from B to A, its components would be [−3,−4] but its magnitude and orientation would be the same.

It is obvious that an infinite number of vectors can have the same components [3,4], since there are an infinite number of pairs of points in the plane with x- and y-coordinates whose respective differences are 3 and 4. All these vectors have the same magnitude and direction, being parallel to one another, and are considered equal. Thus, any vector with components a and b can be considered as equal to the vector [a,b] directed from the origin (0,0) to the point (a,b). The concept of a vector can be extended to three or more dimensions.

Addition and Multiplication of Vectors

The addition, or composition, of two vectors can be accomplished either algebraically or graphically. For example, to add the two vectors U [−3,1] and V [5,2], one can add their corresponding components to find the resultant vector R [2,3], or one can graph U and V on a set of coordinate axes and complete the parallelogram formed with U and V as adjacent sides to obtain R as the diagonal from the common vertex of U and V.

Two different kinds of multiplication are defined for vectors in three dimensions. The scalar, or dot, product of two vectors, A and B, is a scalar, or quantity that has a magnitude but no direction, rather than a vector, and is equal to the product of the magnitudes of A and B and the cosine of the angle θ between them, or A∏B=|A| |B| cos θ. The vector, or cross, product of A and B is a vector, A×B, whose magnitude is equal to |A| |B| sin θ and whose orientation is perpendicular to both A and B and pointing in the direction in which a right-hand screw would advance if turned from A to B through the angle θ. The vector product is an example of a kind of multiplication that does not follow the commutative law, since A×B=−B×A.

Vector Analysis and Vector Space

The components of a vector need not be constants but can also be variables and functions of variables. For example, the position of a body moving through space can be described by a vector whose x,y, and z components are each functions of time. The methods of the calculus may be applied to such vector functions, leading to the branch of mathematics known as vector analysis.

The more general extension of vectors leads to the concept of a vector space. A vector space is a set of elements, A,B,C,…, called vectors, for which the operations of addition of vectors and multiplication of a vector by a scalar are defined and which satisfies ten axioms relating to such properties as closure under both operations, associativity, commutativity, and existence of a zero vector, an additive inverse (negative of a vector), and a unit scalar.

Bibliography

See P. Gustyatnikov and S. Reznichenko, Vector Algebra (1988); J. E. Marsden and A. Tromba, Vector Calculus (1988).

The Columbia Encyclopedia, Sixth Edition Copyright© 2004, Columbia University Press. Licensed from Lernout & Hauspie Speech Products N.V. All rights reserved.

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