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cross - CROSS PRODUCT HOMEWORK 11.4 8 18 38 48 54 Math21a O...

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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( α ) 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( α ). AREA. The length | vectorv × vectorw | is the area of the parallelogram spanned by vectorv and vectorw . Proof. Because | vectorw | sin( α ) is the height of the parallelogram with base length | vectorv | , the area is | vectorv || vectorw | sin( α ) which is by the above formula equal to | vectorv × vectorw | . EXAMPLE. If vectorv = ( a, 0 , 0) and vectorw = ( b cos( α ) , b sin( α ) , 0), then vectorv × vectorw = (0 , 0 , ab sin( α )) which has length | ab sin( α ) | . ZERO CROSS PRODUCT. We see that vectorv × vectorw is zero if vectorv and vectorw are parallel . ORIENTATION. The vectors vectorv, vectorw and vectorv × vectorw form a right handed coordinate system . The right hand rule is: if the first vector vectorv is the thumb, the second vector vectorw is the pointing finger then vectorv × vectorw is the third middle finger of the right hand.
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