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Unformatted text preview: Thevalingam, Donald – Homework 11 – Due: Dec 15 2006, midnight – Inst: Eslami 1 This printout should have 17 questions. Multiplechoice 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 θ 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 θ 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 θ AB = ~ A · ~ B k ~ A kk ~ B k = . 408079 (6) and therefore θ 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 ˆ ı + b ˆ and ~ N = ( c,d ) = c ˆ ı + d ˆ , where a = 4, b = 4, c = 3, and d = 3. a and c represent the xdisplacement and b and d represent the y displacement in a Cartesian xy coordinate system. Note: ˆ ı and ˆ 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 ˆ ı + d ˆ ) · ( c ˆ ı + d ˆ ) = c 2 (ˆ ı · ˆ ı ) + dc (ˆ · ˆ ı ) + cd (ˆ ı · ˆ ) + d 2 (ˆ · ˆ ) = c 2 + d 2 = (3) 2 + ( 3) 2 = 18 . since ˆ ı ⊥ ˆ , we have ˆ ı · ˆ ı = 1, ˆ · ˆ ı = 0, ˆ ı · ˆ = 0, and ˆ · ˆ = 1....
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This note was uploaded on 03/04/2010 for the course PHY 303K taught by Professor Turner during the Spring '08 term at University of Texas.
 Spring '08
 Turner
 Physics, Work

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