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vector: 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. Addition, or composition, of the vectors U [-3,1] and V [5,2] to form the resultant vector R [2,3].

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.

The Columbia Electronic Encyclopedia, 6th ed. Copyright © 2012, Columbia University Press. All rights reserved.

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