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Week_6-7 - Week 6-7 4.4 Matrix Transformations II...

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Week 6-7 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 d 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
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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 [ a b ] T 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( proj d v ) - v ) = 2 proj d v - v T x y = 2 · 1 a 2 + b 2 a 2 ab ab b 2 - 1 0 0 1 x y = 1 a 2 + b 2 a 2 - b 2 2 ab 2 ab b 2 - a 2 x y Example:
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