SECTION 9.7 The Cross Product 671 3 Know Geometric Properties of the Cross Product Now that we have the algebraic definition of the cross product, and have discussed some of its properties, we consider some geometric properties. THEOREM Algebraic Properties of the Cross Product If u, v, and w are vectors in space and if α is a scalar, then • × = u u 0 (2) • u v v u ( ) × = − × (3) • u v u v u v α α α ( ) ( ) ( ) × = × = × (4) • u v w u v u w ( ) ( ) ( ) × + = × + × (5) Proof We will prove properties (2) and (4) here and leave properties (3) and (5) as exercises (see Problems 60 and 61). To prove property (2), let a b c u i j k. 1 1 1 = + + Then a b c a b c b c b c a c a c a b a b u u i j k i j k i j k 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 × = = − + = − + = To prove property (4), let a b c u i j k 1 1 1 = + + and a b c v i j k. 2 2 2 = + + Then α α α α α ( ) ( ) ( ) [ ] ( ) ( ) ( ) ( ) × = − − − + − = − − − + − b c b c a c a c a b a b b c b c a c a c a b a b u v i j k i j k 1 2 2 1 1 2 2 1 1 2 2 1 1 2 2 1 1 2 2 1 1 2 2 1 (6) Since a b c u i j k, 1 1 1 α α α α = + + we have b c b c a c a c a b a b b c b c a c a c a b a b u v i j k i j k 1 2 2 1 1 2 2 1 1 2 2 1 1 2 2 1 1 2 2 1 1 2 2 1 α α α α α α α α α α ( ) ( ) ( ) ( ) ( ) ( ) ( ) × = − − − + − = − − − + − (7) Based on equations (6) and (7), the first part of property (4) follows.The second part can be proved in like fashion. ■ ↑ Use equation (1). THEOREM Geometric Properties of the Cross Product Let u and v be vectors in space. • u v u v is orthogonal to both and . × (8) • θ × = u v u v sin , (9) where θ is the angle between u and v. • ×u v is the area of the parallelogram ≠ ≠ u 0 v 0 having and as adjacent sides. (10) • × = u v 0 u v ifandonlyif and areparallel. (11)
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