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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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This note was uploaded on 02/23/2010 for the course MATH 221 taught by Professor Staff during the Spring '08 term at Tulane.
 Spring '08
 staff
 Vectors, Dot Product

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