hw4 - This print-out should have 15 questions...

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Unformatted text preview: This print-out should have 15 questions. Multiple-choice questions may continue on the next column or page – find all choices before answering. 001 10.0 points Find the scalar projection of b onto a when b = 2 i + 3 j + k , a = i + 2 j + 2 k . 1. scalar projection = 3 2. scalar projection = 11 3 3. scalar projection = 4 4. scalar projection = 8 3 5. scalar projection = 10 3 correct Explanation: The scalar projection of b onto a is given in terms of the dot product by comp a b = a · b | a | . Now when b = 2 i + 3 j + k , a = i + 2 j + 2 k , we see that a · b = 10 , | a | = radicalBig (1) 2 + (2) 2 + (2) 2 . Consequently, comp a b = 10 3 . keywords: 002 10.0 points Find the vector projection of b onto a when b = ( 4 , − 3 ) , a = ( 1 , − 4 ) . 1. vector projection = 1 ( 1 , − 4 ) 2. vector projection = 17 √ 17 ( 4 , − 3 ) 3. vector projection = 16 17 ( 1 , − 4 ) cor- rect 4. vector projection = 17 √ 17 ( 1 , − 4 ) 5. vector projection = 16 √ 17 ( 4 , − 3 ) 6. vector projection = 16 17 ( 4 , − 3 ) Explanation: The vector projection of b onto a is given in terms of the dot product by proj a b = parenleftBig a · b | a | 2 parenrightBig a . Now when b = ( 4 , − 3 ) , a = ( 1 , − 4 ) , we see that a · b = 16 , | a | = radicalBig (1) 2 + ( − 4) 2 . Consequently, proj a b = 16 17 ( 1 , − 4 ) . keywords: 003 10.0 points The box shown in x y z A B C D is the unit cube having one corner at the origin and the coordinate planes for three of its adjacent faces. Determine the vector projection of −−→ AB onto −→ AC . 1. vector projection = − 1 2 ( j − k ) 2. vector projection = 1 2 ( j − k ) 3. vector projection = − 2 3 ( i + j − k ) 4. vector projection = 1 2 ( i − k ) 5. vector projection = − 1 2 ( i − k ) 6. vector projection = 2 3 ( i + j − k ) correct Explanation: The vector projection of a vector b onto a vector a is given in terms of the dot product by proj a b = parenleftBig a · b | a | 2 parenrightBig a . On the other hand, since the unit cube has side-length 1, A = (0 , , 1) , B = (1 , , 0) , while C = (1 , 1 , 0). In this case −→ AC is a directed line segment determining the vector a = ( 1 , 1 , − 1 ) = i + j − k , , while −−→ AB determines the vector b = ( 1 , , − 1 ) = i − k . For these choices of a and b , a · b = 2 , | a | 2 = 3 . Consequently, the vector projection of −−→ AB onto −→ AC is given by proj a b = 2 3 ( i + j − k ) . keywords: vector projection, dot product, unit cube, component, 004 10.0 points Find the cross product of the vectors a = (− 3 , − 1 , − 2 ) , b = (− 2 , 2 , − 2 ) ....
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This note was uploaded on 09/28/2008 for the course M 408M taught by Professor Gilbert during the Fall '07 term at University of Texas.

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hw4 - This print-out should have 15 questions...

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