calassignment 4-solutions

calassignment 4-solutions - miller(zdm77 assignment 4...

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miller (zdm77) – assignment 4 – luecke – (58600) 1 This print-out should have 15 questions. Multiple-choice questions may continue on the next column or page – fnd all choices beFore answering. 001 10.0 points ±ind the angle between the vectors a = a 1 , 2 3 A , b = a− 3 , 7 A . 1. angle = π 6 correct 2. angle = 3 π 4 3. angle = π 4 4. angle = 5 π 6 5. angle = 2 π 3 6. angle = π 3 Explanation: Since the dot product oF vectors a and b can be written as a . b = | a || b | cos θ , 0 θ π, where θ is the angle between the vectors, we see that cos θ = a . b | a b | , 0 θ π . But For the given vectors, a · b = (1)( 3) + (2 3)(7) = 13 3 , while | a | = 13 , | b | = 52 . Consequently, cos θ = 13 3 13 · 2 13 = 3 2 where 0 θ π . Thus angle = π 6 . keywords: vectors, dot product, angle be- tween vectors 002 10.0 points A triangle Δ PQR in 3-space has vertices P (1 , 6 , 3) , Q (2 , 3 , 1) , R (6 , 5 , 2) . Use vectors to decide which one oF the Follow- ing properties the triangle has. 1. right-angled at R 2. not right-angled at P, Q, or R correct 3. right-angled at P 4. right-angled at Q Explanation: Vectors a and b are perpendicular when a · b = 0. Thus Δ will be (1) right-angled at P when −−→ QP · −→ RP = 0, (2) right-angled at Q when PQ · RQ = 0, (3) right-angled at R when PR · QR = 0. But For the vertices P (1 , 6 , 3) , Q (2 , 3 , 1) , R (6 , 5 , 2) we see that = a 1 , 3 , 2 A , QR = a 4 , 2 , 1 A , while RP = 5 , 1 , 1 A . Thus QP · RP = 10 , · RQ = 4 , and · QR = 17 . Consequently, Δ is not right-angled at P, Q, or R .
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miller (zdm77) – assignment 4 – luecke – (58600) 2 keywords: vectors, dot product, right trian- gle, perpendicular, 003 10.0 points Find the vector projection of b onto a when b = a− 2 , 4 A , a = a 1 , 4 A . 1. vector projection = 1 a 1 , 4 A 2. vector projection = 18 17 2 , 4 A 3. vector projection = 17 17 a 1 , 4 A 4. vector projection = 17 17 2 , 4 A 5. vector projection = 18 17 2 , 4 A 6. vector projection = 18 17 a 1 , 4 A cor- rect Explanation: The vector projection of b onto a is given in terms of the dot product by proj a b = p a · b | a | 2 P a . Now when b = 2 , 4 A , a = a 1 , 4 A , we see that a · b = 18 , | a | = r (1) 2 + ( 4) 2 . Consequently, proj a b = 18 17 a 1 , 4 A . keywords: 004 10.0 points Find the vector projection of b onto a when b = i + 3 j k , a = 2 i + j + 3 k . 1. vector projection = 3 14 ( 2 i + j + 3 k ) 2. vector projection = 2 7 ( i + 3 j k ) 3. vector projection = 3 14 ( i + 3 j k ) 4. vector projection = 1 7 ( i + 3 j k ) 5. vector projection = 1 7 ( 2 i + j + 3 k ) correct 6. vector projection = 2 7 ( 2 i + j + 3 k ) Explanation: The vector projection of b onto a is given in terms of the dot product by proj a b = p a · b | a | 2 P a .
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calassignment 4-solutions - miller(zdm77 assignment 4...

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