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ch12s4 - 12.4 The Cross Product The cross product is a...

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12.4 The Cross Product The cross product is a vector, sometimes called the vector product. Cross products are only found with 3-dimensional vectors. Cross Product 1, 2 3 1 2 3 2 3 3 2 3 1 1 3 1 2 2 1 , , , , , a a a b b b a b a b a b a b a b a b = = × = a b a b Determinants can be used to find this product. The TI-85 and TI-86 will also perform this operation. Determinants and Cross Product 1 2 3 1 2 3 2 3 1 3 1 2 2 3 1 3 1 2 1 2 3 1 2 3 , , , , a b ad bc c d a a a b b b a a a a a a b b b b b b a a a b b b = = = × = + × = a b a b i j k i j k a b Example: Find the cross product for 1,2, 3 5, 1, 2 = = a b . Theorems, etc. (a) The cross product is orthogonal to both a and b. (b) If θ is the angle between a and b then sin θ × = a b a b . (c) Two nonzero vectors a and b are parallel if and only if the cross product is 0.
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