hw10sol - Homework 10 Solutions 1. Use polar coordinates to...

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Homework 10 Solutions 1. Use polar coordinates to prove that the transformation given by the matrix " cos θ - sin θ sin θ cos θ # is ρ ~ 0 (i.e. a rotation about the origin by θ ). Solution: Let ( r cos α,r sin α ) be an arbitrary point in R 2 . Now we have " cos θ - sin θ sin θ cos θ #" r cos α r sin α # = " r cos α cos θ - r sin α sin θ r cos α sin θ + r sin α cos θ # = " r cos( α + θ ) r sin( α + θ ) # It follows form this that the point with polar coordinates ( r,α ) goes to the point with polar coordinates ( r,α + θ ) so this matrix is rotating about the origin by an angle of θ , as claimed. 2. What transformation is given by the matrix " cos θ sin θ sin θ - cos θ # ? Solution: Again we let ( r cos α,r sin α ) be an arbitrary point in R 2 . Now we have " cos θ sin θ sin θ - cos θ #" r cos α r sin α # = " r cos α cos θ + r sin α sin θ r cos α sin θ - r sin α cos θ # = " r cos( θ - α ) r sin( θ - α ) # So, the point with polar coordinates ( r,α ) goes to the point with polar coordinates ( r,θ - α ).
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hw10sol - Homework 10 Solutions 1. Use polar coordinates to...

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