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Unformatted text preview: 12.4 The Cross Product VECTORS AND THE GEOMETRY OF SPACE In this section, we will learn about: Cross products of vectors and their applications. THE CROSS PRODUCT The cross product a x b of two vectors a and b , unlike the dot product, is a vector. For this reason, it is also called the vector product. s Note that a x b is defined only when a and b are threedimensional (3D) vectors. If a = a 1 , a 2 , a 3 = a 1 i + a 2 j + a 3 k and b = b 1 , b 2 , b 3 = b 1 i + b 2 j + b 3 k , then the cross product of a and b is the vector a x b = a 2 b 3 a 3 b 2 , a 3 b 1 a 1 b 3 , a 1 b 2 a 2 b 1 A determinant of order 2 is defined by: 2 3 1 3 1 2 2 3 1 3 1 2 a a a a a a b b b b b b = + a b i j k a b ad bc c d = 1 2 3 1 2 3 a a a b b b = i j k a b The vector a x b is orthogonal to both a and b . CROSS PRODUCT Theorem 2 3 1 3 1 2 2 3 ( ) a a a a a a a a  + a b a Proof A similar computation shows that ( a x b ) b = 0 s Therefore, a x b is orthogonal to both...
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 Spring '08
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 Vectors, Dot Product

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