Gilbert_Hmwk09sol - moseley (cmm3869) HW09 Gilbert (56380)...

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Unformatted text preview: moseley (cmm3869) HW09 Gilbert (56380) 1 This print-out should have 23 questions. Multiple-choice questions may continue on the next column or page find all choices before answering. 001 10.0 points Determine the dot product of the vectors a = i + 2 j + k , b = i 3 j + k . 1. a b = 4 correct 2. a b = 10 3. a b = 2 4. a b = 6 5. a b = 8 Explanation: The dot product, a b , of vectors a = a 1 i + a 2 j + a 3 k , b = b 1 i + b 2 j + b 3 k is defined by a b = a 1 b 1 + a 2 b 2 + a 3 b 3 . Consequently, when a = i + 2 j + k , b = i 3 j + k , we see that a b = 4 . 002 10.0 points Determine the dot product of vectors a , b when | a | = 5 , | b | = 3 and the angle between a and b is / 3. 1. a b = 8 2. a b = 7 3. a b = 9 4. a b = 15 2 correct 5. a b = 17 2 Explanation: The dot product of vectors a , b is defined in coordinate-free form by a b = | a || b | cos where is the angle between a and b . For the given vectors, therefore, a b = 15 cos 3 = 15 2 . 003 10.0 points Which of the following statements are true for all vectors a , b ? A. | a b | 2 = | a | 2 + 2 a b + | b | 2 , B. | a b | = | a || b | = a bardbl b , C. a b = 0 = a = 0 or b = 0. 1. A only 2. C only 3. A and B only 4. A and C only 5. B and C only 6. none of them 7. all of them 8. B only correct Explanation: If is the angle between a and b , then a b = | a || b | cos . moseley (cmm3869) HW09 Gilbert (56380) 2 A. FALSE: since | a | 2 = a a , | a b | 2 = ( a b ) ( a b ) = | a | 2 a b b a + | b | 2 = | a | 2 2 a b + | b | 2 because a b = b a . B. TRUE: since | a b | = | a || b | = | cos | = 1 , it follows that = 0 or , in which case a bardbl b . C. FALSE: if a b , then = / 2. But then cos = 0. So a b = 0 when a b , as well as when a = 0 or b = 0. keywords: 004 10.0 points Find the angle between the vectors a = ( 2 3 , 1 ) , b = ( 3 3 , 5 ) . 1. angle = 2 3 2. angle = 3 4 3. angle = 3 correct 4. angle = 6 5. angle = 5 6 6. angle = 4 Explanation: Since the dot product of vectors a and b can be written as a . b = | a || b | cos , , where is the angle between the vectors, we see that cos = a . b | a || b | , . But for the given vectors, a b = (2 3)(3 3) + ( 1)(5) = 13 , while | a | = 13 , | b | = 52 . Consequently, cos = 13 13 2 13 = 1 2 where 0 . Thus angle = 3 . 005 10.0 points For which positive value of x are the vectors ( 24 x, 2 , 1 ) , ( 2 , 5 x 2 , 10 ) orthogonal?...
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Gilbert_Hmwk09sol - moseley (cmm3869) HW09 Gilbert (56380)...

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