Gilbert_Hmwk09sol

# Gilbert_Hmwk09sol - moseley(cmm3869 – HW09 – Gilbert...

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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 π/ 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 θ where θ is the angle between a and b . For the given vectors, therefore, a · b = 15 cos π 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 θ is the angle between a and b , then a · b = | a || b | cos θ . 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 θ | = 1 , it follows that θ = 0 or π , in which case a bardbl b . C. FALSE: if a ⊥ b , then θ = π/ 2. But then cos θ = 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 π 3 2. angle = 3 π 4 3. angle = π 3 correct 4. angle = π 6 5. angle = 5 π 6 6. angle = π 4 Explanation: Since the dot product of vectors a and b can be written as a . b = | a || b | cos θ , ≤ θ ≤ π, where θ is the angle between the vectors, we see that cos θ = a . b | a || b | , ≤ θ ≤ π . But for the given vectors, a · b = (2 √ 3)(3 √ 3) + ( − 1)(5) = 13 , while | a | = √ 13 , | b | = √ 52 . Consequently, cos θ = 13 √ 13 · 2 √ 13 = 1 2 where 0 ≤ θ ≤ π . Thus angle = π 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...

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