Exam 1B -- Solution

# Exam 1B -- Solution - MAC2313 Exam 1 Version B Solution 1....

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Unformatted text preview: MAC2313 Exam 1 Version B Solution 1. Find a unit vector orthogonal to the vectors ~a = h- 2 , 1 , 4 i and ~ b = h- 1 , , 3 i . (10 points) Solution: There are two unit vectors that are orthogonal to ~a and ~ b . Let ~v = ~a ~ b = h 3 , 2 , 1 i . ~v is not a unit vector since || ~v || = 14. The 2 unit vectors orthogonal to ~a and ~ b are ~v || ~v || = 3 14 , 2 14 , 1 14- ~v || ~v || =- 3 14 ,- 2 14 ,- 1 14 . 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 ) = sin( t ) , csc( t ) , cos 2 ( t ) lies. (Sketch the surfaces on two separate graphs). (15 points) Solution: Three surfaces can be quickly identified using relations between two component...
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## This note was uploaded on 12/15/2011 for the course MAC 2313 taught by Professor Keeran during the Spring '08 term at University of Florida.

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Exam 1B -- Solution - MAC2313 Exam 1 Version B Solution 1....

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