SOL11 - Thevalingam, Donald Homework 11 Due: Dec 15 2006,...

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Unformatted text preview: Thevalingam, Donald Homework 11 Due: Dec 15 2006, midnight Inst: Eslami 1 This print-out should have 17 questions. Multiple-choice questions may continue on the next column or page find all choices before answering. The due time is Central time. 001 (part 1 of 1) 10 points Vector ~ A has components A x =- 7 . 7 , A y = 8 . 5 , A z = 4 , while vector ~ B has components B x = 3 . 7 , B y =- 4 . 8 , B z = 6 . 4 . What is the angle AB between these vec- tors? (Answer between 0 and 180 .) Correct answer: 114 . 084 . Explanation: Note: The magnitude of vector ~ X is k X k . Consider two formulae for the scalar prod- uct ~ A ~ B of two vectors: ~ A ~ B = A x B x + A y B y + A z B z (1) in terms of the two vectors components, and also ~ A ~ B = k ~ A kk ~ B k cos AB (2) in term of their magnitudes and the angle be- tween them. Given the data, we immediately calculate k ~ A k = q A 2 x + A 2 y + A 2 z = 12 . 1466 , (3) k ~ B k = q B 2 x + B 2 y + B 2 z = 8 . 81419 , (4) and using eq. (1), ~ A ~ B =- 43 . 69 . (5) Hence, according to eq. (2), cos AB = ~ A ~ B k ~ A kk ~ B k =- . 408079 (6) and therefore AB = arccos(- . 408079) = 114 . 084 . (7) Two vectors always lie in a plane. When these two vectors are plotted in this plane, we have A B 1 1 4 . 8 4 keywords: 002 (part 1 of 2) 10 points Consider the two vectors ~ M = ( a,b ) = a + b and ~ N = ( c,d ) = c + d , where a = 4, b = 4, c = 3, and d =- 3. a and c represent the x-displacement and b and d represent the y- displacement in a Cartesian xy co-ordinate system. Note: and represent unit vectors ( i.e. vectors of length 1) in the x and y directions, respectively. What is the value of the scalar product ~ N ~ N ? Correct answer: 18 . Explanation: Take the scalar products of the x- and y- displacement of ~ N and ~ N individually ~ N ~ N = ( c + d ) ( c + d ) = c 2 ( ) + dc ( ) + cd ( ) + d 2 ( ) = c 2 + d 2 = (3) 2 + (- 3) 2 = 18 . since , we have = 1, = 0, = 0, and = 1....
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SOL11 - Thevalingam, Donald Homework 11 Due: Dec 15 2006,...

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