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Vector Product

Theorem 3 raises the following problem about vectors in space. Given two vectors A and B which are neither zero nor parallel, find a third vector C which is perpendicular to both A and B. For example, if A is i and B is j, then the vector k is perpendicular to both i and j. So is any scalar multiple of k. In general it is not easy to see how to find a vector perpendicular to both A and B. In fact, to solve the problem we need a new kind of product of vectors, the vector product

A × B.

DEFINITION

Given two vectors

A = a1i + a2j + a3k, B = b1i + b2j + b3k

in space, the vector product (or cross product) is the new vector

A × B = (a2b3 - a3b2)i + (a3b1 - a1b3)j + (a1b2 - a2b1)k.

This definition can be remembered by writing down the determinant

10_vectors-155.gif

The positive and negative terms of A × B are the products of the diagonals shown in Figure 10.4.5.

10_vectors-156.gif

Figure 10.4.5

Example 9: Finding a Vector Product


Last Update: 2006-11-05