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Exam 1A -- Solution - MAC2313 Exam 1 Version A Solution 1...

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MAC2313 Exam 1 Version A Solution 1. Find a unit vector orthogonal to the vectors ~a = h 3 , 1 , - 1 i and ~ b = h- 1 , 0 , 3 i . (10 points) Solution: There are two unit vectors that are orthogonal to ~a and ~ b . Let ~v = ~a × ~ b = h 3 , - 8 , 1 i . ~v is not a unit vector since || ~v || = 74. The 2 unit vectors orthogonal to ~a and ~ b are ~v || ~v || = 3 74 , - 8 74 , 1 74 - ~v || ~v || = - 3 74 , 8 74 , - 1 74 . 2. Let ~a, ~ b be vectors in R 3 and let orth ~a ( ~ b ) = ~ b - proj ~a ( ~ b ) be the orthogonal projection of ~ b onto ~a . Prove that orth ~a ( ~ b ) is orthogonal to ~a . (15 points) Solution: To show that two vectors are orthogonal, we must show that the dot product is 0. Remember that ~a ~a = || ~a || 2 , ~a orth ~a ( ~ b ) = ~a ( ~ b - proj ~a ( ~ b )) = ~a ~ b - ~a proj ~a ( ~ b ) = ~a ~ b - ~a ~a ~ b || ~a || 2 ! ~a = ~a ~ b - ~a ~ b || ~a || 2 ! ~a ~a = ~a ~ b - || ~a || 2 || ~a || 2 ! ( ~a ~ b ) = 0 . 3. Find the equation and sketch the graph of 2 surfaces in which the curve ~ r ( t ) = cos( t ) , sec( t ) , sin 2 ( t ) lies. (Sketch the surfaces on two separate graphs). (15 points) Solution: Three surfaces can be quickly identified using relations between two component functions of ~ r .
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