cross - CROSS PRODUCT Math21a, O. Knill HOMEWORK: 11.4: 8,...

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Unformatted text preview: CROSS PRODUCT Math21a, O. Knill HOMEWORK: 11.4: 8, 18, 38, 48, 54 CROSS PRODUCT. The cross product of two vectors vectorv = ( v 1 ,v 2 ,v 3 ) and vectorw = ( w 1 ,w 2 ,w 3 ) is defined as the vector vectorv vectorw = ( v 2 w 3 v 3 w 2 ,v 3 w 1 v 1 w 3 ,v 1 w 2 v 2 w 1 ) . To compute it, look at the determinant: i j k v 1 v 2 v 3 w 1 w 2 w 3 = i v 2 v 3 w 2 w 3 j v 1 v 3 w 1 w 3 + k v 1 v 2 w 1 w 2 = i ( v 2 w 3 v 3 w 2 ) j ( v 1 w 3 v 3 w 1 ) + k ( v 1 w 2 v 2 w 1 ). DIRECTION OF vectorv vectorw : vectorv vectorw is orthogonal to vectorv and orthogonal to vectorw . Proof. Check that vectorv ( vectorv vectorw ) = 0. LENGTH: | vectorv vectorw | = | vectorv || vectorw | sin( ) Proof. The identity | vectorv vectorw | 2 = | vectorv | 2 | vectorw | 2 ( vectorv vectorw ) 2 can be proven by direct computation. Now, | vectorv vectorw | = | vectorv || vectorw | cos( ). AREA. The length | vectorv vectorw | is the area of the parallelogram spanned by vectorv and vectorw . Proof. Because | vectorw | sin( ) is the height of the parallelogram with base length | vectorv | , the area is | vectorv || vectorw | sin( ) which is by the above formula equal to | vectorv vectorw | . EXAMPLE. If vectorv = ( a, , 0) and vectorw = ( b cos( ) ,b sin( ) , 0), then vectorv vectorw = (0 , ,ab sin( )) which has length | ab sin( ) | . ZERO CROSS PRODUCT. We see that vectorv vectorw is zero if vectorv and vectorw are parallel ....
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This note was uploaded on 03/29/2008 for the course MATH 251 taught by Professor Skrypka during the Spring '08 term at Texas A&M.

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