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Unformatted text preview: Fact Sheet for Final Exam Vector parametrization of the line through ( x ,y ,z ) in the direction v = ( a,b,c ): r ( t ) = ( x ,y ,z ) + t ( a,b,c ). The dot product of v = ( a 1 ,b 1 ,c 1 ) and w = ( a 2 ,b 2 ,c 2 ): v w = a 1 ,a 2 + b 1 b 2 + c 1 c 2 . v w =  v  w  cos , with the angle between v and w . The orthogonal projection of u on v is proj v ( u ) = u v v v v . The cross product of v = ( a 1 ,b 1 ,c 1 ) and w = ( a 2 ,b 2 ,c 2 ): v w = i j k a 1 b 1 c 1 a 2 b 2 c 2 = b 1 c 1 b 2 c 2 i a 1 c 1 a 2 c 2 j + a 1 b 1 a 2 b 2 k The area of the parallelogram spanned by v and w is  v w  . The volume of the parallelepiped spanned by u , v and w is  u ( v w )  . An equation of the plane through ( x ,y ,z ) with normal vector n = ( a,b,c ) can be written in vector form as n ( x x ,y y ,z z ) = 0; in scalar form as ax + by + cz = ax + bx + cz . Let f be a function from R n to R , then the gradient of f is f = ( f x 1 , , f x n )....
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This note was uploaded on 04/21/2010 for the course MATH 20E taught by Professor Enright during the Winter '07 term at UCSD.
 Winter '07
 Enright
 Math, Vector Calculus, Dot Product

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