vectcross

# 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&amp;nitions Unlike the dot product, the cross product is only de&amp;ned for 3-D vectors. In this section, when we use the word vector, we will mean 3-D vector. De&amp;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 &amp; ~v , is de&amp;ned to be @ u x u y u z 1 A &amp; @ 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 &amp;rst quickly review what they are. De&amp;nition 45 We only give the de&amp;nition of the determinant of a 2 &amp; 2 and a 3 &amp; 3 matrix. 1. The determinant of a 2 &amp; 2 matrix &amp; a b c d , denoted by a b c d is de&amp;ned to be a b c d = ad bc 2. The determinant of a 3 &amp; 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&amp;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 &amp; &amp; &amp; &amp; &amp; &amp; 1 2 3 3 1 1 4 7 2 &amp; &amp; &amp; &amp; &amp; &amp; &amp; &amp; &amp; &amp; &amp; &amp; 1 2 3 3 1 1 4 7 2 &amp; &amp; &amp; &amp; &amp; &amp; = 1 &amp; &amp; &amp; &amp; 1 1 7 2 &amp; &amp; &amp; &amp; &amp; 2 &amp; &amp; &amp; &amp; 3 1 4 2 &amp; &amp; &amp; &amp; + 3 &amp; &amp; &amp; &amp; 3 1 4 7 &amp; &amp; &amp; &amp; = (1)(2 &amp; 7) &amp; 2(6 &amp; 4) + 3(21 &amp; 4) = &amp; 5 &amp; 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 = &amp; &amp; &amp; &amp; &amp; &amp; &amp;! i &amp;! j &amp;! k u x u y u z v x v y v z &amp; &amp; &amp; &amp; &amp; &amp; 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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