vectcross - 24 CHAPTER 1. VECTORS AND THE GEOMETRY OF SPACE...

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Unformatted text preview: 24 CHAPTER 1. VECTORS AND THE GEOMETRY OF SPACE 1.4 Cross Product 1.4.1 De&nitions Unlike the dot product, the cross product is only de&ned for 3-D vectors. In this section, when we use the word vector, we will mean 3-D vector. De&nition 44 (cross product) The cross product also called vector prod- uct of two vectors ~u = h u x ;u y ;u z i and ~v = h v x ;v y ;v z i , denoted ~u & ~v , is de&ned to be @ u x u y u z 1 A & @ v x v y v z 1 A = @ u y v z u z v y u z v x u x v z u x v y u y v x 1 A Thus, the cross product of two 3-D vectors is also a 3-D vector. This formula is not easy to remember. However, if you know about matrices and the determinant of a matrix, the cross product can be expressed in term of them. Let us &rst quickly review what they are. De&nition 45 We only give the de&nition of the determinant of a 2 & 2 and a 3 & 3 matrix. 1. The determinant of a 2 & 2 matrix & a b c d , denoted by a b c d is de&ned to be a b c d = ad bc 2. The determinant of a 3 & 3 matrix 2 4 a 1 a 2 a 3 b 1 b 2 b 3 c 1 c 2 c 3 3 5 denoted by a 1 a 2 a 3 b 1 b 2 b 3 c 1 c 2 c 3 is de&ned to be a 1 a 2 a 3 b 1 b 2 b 3 c 1 c 2 c 3 = a 1 b 2 b 3 c 2 c 3 a 2 b 1 b 3 c 1 c 3 + a 3 b 1 b 2 c 1 c 2 = a 1 ( b 2 c 3 c 2 b 3 ) a 2 ( b 1 c 3 c 1 b 3 ) + a 3 ( b 1 c 2 c 1 b 2 ) Example 46 Find 1 2 7 3 1 2 7 3 = (1)(3) (7)(2) = 3 14 = 11 1.4. CROSS PRODUCT 25 Example 47 Find & & & & & & 1 2 3 3 1 1 4 7 2 & & & & & & & & & & & & 1 2 3 3 1 1 4 7 2 & & & & & & = 1 & & & & 1 1 7 2 & & & & & 2 & & & & 3 1 4 2 & & & & + 3 & & & & 3 1 4 7 & & & & = (1)(2 & 7) & 2(6 & 4) + 3(21 & 4) = & 5 & 4 + 51 = 42 Proposition 48 If ~u = h u x ;u y ;u z i and ~v = h v x ;v y ;v z i then ~u ~v = & & & & & & &! i &! j &! k u x u y u z v x v y v z & & & & & & Which makes it much easier to remember....
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vectcross - 24 CHAPTER 1. VECTORS AND THE GEOMETRY OF SPACE...

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