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Unformatted text preview: To determine the direction of u v we have the right hand rule; Fingers point the first vector, Palm points the second vector, tumb gives the direction of the cross product. Theorem 3.5.1 (Relationship involving Cross Product and Dot Prod uct:) If u,v and w are vectors in 3space, then (a) u ( u v ) = 0 (that is u v and u are orthogonal) (b) v ( u v ) = 0 (that is u v and v are orthogonal) (c)  u v  2 =  u  2  v  2 ( u v ) 2 (Lagranges Identity) (d) u ( v w ) = ( u w ) v ( u v ) w (e) ( u v ) w = ( u w ) v ( v w ) u Proof: Write u = ( u 1 ,u 2 ,u 3 ) , v = ( v 1 ,v 2 ,v 3 ) and apply the definitions of dot prod uct and cross product. Importance: From part (a) and (b) we see that if one needs to find a vector that is orthogonal to both vectors u and v then u v will do the work. We will see that this property can be used to write a plane equation. Example: Let u = (1 , , 2) ,v = ( 1 , 1 , 3) then u v = i j k 1 0 2 1 1 3 = 2 i 5 j + k = ( 2 , 5 , 1) ( u v ) u = ( 2 , 5 , 1) (1 , , 2) = 0 , ( u v ) v = ( 2 , 5 , 1) ( 1 , 1 , 3) = 0 . Theorem 3.5.2 (Properties of Cross Product): If u,v and w are any vectors in 3space and k is any scalar, then (a) u v = ( v u ) (b) u ( v + w ) = ( u v ) + ( u w ) (c) ( u + v ) w = ( u w ) + ( v w ) (d) k ( u v ) = ( ku ) v = u ( kv ) 1 (e) u 0 = 0 u = 0 (f) u u = 0 Geometric Interpretation of Cross Product and Determinants: Theorem 3.5.3 (Area of a Parallelogram) If u and v are vectors in 3space, then  u v  is equal to the area of the parallelogram determined by u and v. Why  u v  =  u  v  sin ( ) ? From Theorem 3.5.1 we have Lagranges Identity  u v  2 =  u  2  v  2 ( u v ) 2 and we have cos ( ) = u v  u  v  u v =  u  v  cos ( ) . Then  u v  2 =  u  2  v  2 ( u v ) 2 =  u  2  v  2  u  2  v  2 cos 2 ( ) =  u  2  v  2 (1 cos 2 ( )) =  u  2  v  2 sin 2 ( )  u v  =  u ...
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This note was uploaded on 04/27/2011 for the course ENGR 130 taught by Professor Zhang during the Spring '11 term at University of Alberta.
 Spring '11
 Zhang

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