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Unformatted text preview: Week 67 4.4 Matrix Transformations II Projection of a Vector ~v on a Line Through the Origin with Direction Vector ~ d = [ a b c ] T . We will let ~v = [ x y z ] T . Recall that proj ~ d ~v = ~v ~ d k ~ d k 2 ~ d = ax + by + cz a 2 + b 2 + c 2 a b c . w 1 = ax + by + cz a 2 + b 2 + c 2 a = a 2 x + aby + acz a 2 + b 2 + c 2 w 2 = ax + by + cz a 2 + b 2 + c 2 c = acx + bcy + c 2 z a 2 + b 2 + c 2 w 3 = ax + by + cz a 2 + b 2 + c 2 c = acx + bcy + c 2 z a 2 + b 2 + c 2 Then the standard matrix for this transformation is 1 a 2 + b 2 + c 2 a 2 ab ac ab b 2 bc ac bc c 2 Projection of a Vector ~v onto a Plane Through the Origin with Normal Vector ~n = [ a b c ] T . Here we want the vector ~v proj ~n ~v or if we let ~v = [ x y z ] T , then we have ~v proj ~n ~v = x y z  1 a 2 + b 2 + c 2 a 2 ab ac ab b 2 bc ac bc c 2 x y z = 1 0 0 0 1 0 0 0 1  1 a 2 + b 2 + c 2 a 2 ab ac ab b 2 bc ac bc c 2 x y z = 1 a 2 + b 2 + c 2 b 2 + c 2 ab ac ab a 2 + c 2 bc ac bc a 2 + b 2 x y z Reflection of a Vector ~v in a Line Through the Origin with Direction Vector ~ d = [ a b c ] T Standard Matrix is 1 a 2 + b 2 + c 2 a 2 ab ac ab b 2 bc ac bc c 2 . Reflection of a Vector ~v in a Plane Through the Origin with Normal Vector ~n = [ a b c ] T Standard Matrix is 1 a 2 + b 2 + c 2 b 2 + c 2 a 2 2 ab 2 ac 2 ab a 2 + c 2 b 2 2 bc 2 ac 2 bc a 2 + b 2 c 2 . 1 Projection of a vector ~v onto a Line Through the Origin with Direction Vector ~ d = [ a b ] T If we let ~v = [ x y ] T , then we have T ( ~v ) = proj ~ d ~v T x y = [ x y ] T [ a b ] T k [ a b ] T k 2 a b = 1 a 2 + b 2 ( ax + by ) a b = 1 a 2 + b 2 a 2 x + bay abx + b 2 y = 1 a 2 + b 2 a 2 ab ab b 2 x y Reflection of a vector ~v about a Line Through the Origin with Direction Vector ~ d = [ a b ] T If we let ~v = [ x y ] T , then we have T ( ~v ) = ~v + 2(...
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This note was uploaded on 08/05/2008 for the course MATH 115 taught by Professor Dunbar during the Fall '07 term at Waterloo.
 Fall '07
 DUNBAR
 Transformations

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