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