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Math 192 Final Exam Cheat Sheet

# Math 192 Final Exam Cheat Sheet - cos u v | u | v | projv u...

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| || | cos v u v u = θ v v v u u v = 2 | | proj θ cos | || | v u v u = θ sin | || | v u v u = × v u × Volume of Parallelogram w v u × ) ( Triple Scalar Product (Volume of Parallelepiped) v r r t t + = 0 ) ( Vector equation of a line Distance from a point to a line | | | | v v × = PS d Given: normal: Ck Bj Ai + + = n u ) , , ( 0 0 0 0 z y x P = Equation of a Plane: 0 0 0 Cz By Ax D + + = D Cz By Ax = + + Distance from a point to a plane - n vector normal to plane - S point - P any point on plane | | n n PS d r r = | | ) ( ) ( 2 1 2 1 1 2 v v v v r r × × - = d Distance between two skew lines Vector as a sum of vectors parallel and perpendicular to v ) ( u u u v v proj u proj - + = Tangent Vector | ) ( | ) ( ) ( t v t v t T = Chain Rule dt dz dz df dt dy dy df dt dx dx df dt dw + + = y x F F dx dy - = Implicit Form Directional Derivative θ cos | || | 0 0 u u P P u f f D = = Tangent Plane ) ) , , ( ( 0 P at z y x f 0 ) )( ( ) )( ( ) )( ( 0 0 0 0 0 0 = - + - + - z z P f y y P f x x P f z y x 0 ) ( ) )( ( ) )( ( 0 0 0 0 0 = - - - + - z z y y P f x x P f y x ) ) , ( ( 0 P at y x f Tangent Plane Estimating Change ds f df P ) ( 0 u = Linearization ) )( , ( ) )( , ( ) , ( ) , ( 0 0 0 0 0 0 0 0 y y y x f x x y x f y x f y x L y x - + - + =

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