HW24 - Pickering, Tracey Homework 24 Due: Mar 31 2006, noon...

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Pickering, Tracey – Homework 24 – Due: Mar 31 2006, noon – Inst: Drummond 1 This print-out should have 15 questions. Multiple-choice questions may continue on the next column or page – fnd all choices beFore answering. The due time is Central time. 001 (part 1 oF 3) 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 and c represent the x -displacement and b and d represent the y -displacement in a Cartesian xy co-ordinate 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 magnitude oF the vector prod- uct k ~ M × ~ M k ? 1. k ~ M × ~ M k = a 2 - 2 a b + b 2 2. k ~ M × ~ M k = 2 a b 3. k ~ M × ~ M k = a 2 + b 2 4. k ~ M × ~ M k = a 2 - b 2 5. k ~ M × ~ M k = a b 6. k ~ M × ~ M k = p a 2 + b 2 7. k ~ M × ~ M k = a 2 + 2 a b + b 2 8. k ~ M × ~ M k = a + b 9. k ~ M × ~ M k = 0 correct 10. k ~ M × ~ M k = - a b Explanation: The magnitude oF the vector product oF two vectors k ~ A × ~ B k = A B sin θ , where θ is the angle between the two vectors placed with their tails at the same point. When ~ A and ~ B are the same vector, θ = 0 . Since sin 0 = 0, the vector product oF a vector with itselF is zero. 002 (part 2 oF 3) 10 points What is the magnitude oF the vector product k ~ M × ~ N k ? 1. k ~ M × ~ N k = a c + b d 2. k ~ M × ~ N k = a 2 + b 2 + c 2 + d 2 3. k ~ M × ~ N k = a b c d 4. k ~ M × ~ N k = a d - b c correct 5. k ~ M × ~ N k = a b - c d 6. k ~ M × ~ N k = 0 7. k ~ M × ~ N k = a d + b c 8. k ~ M × ~ N k = a b + c d 9. k ~ M × ~ N k = a - b 10. k ~ M × ~ N k = a c - b d Explanation: Take the vector products oF the x - and y - displacement oF ~ M and ~ N individually ~ M × ~ N = ( a ˆ ı + b ˆ ) × ( c ˆ ı + d ˆ ) = a c ı × ˆ ı ) + b c × ˆ ı ) + a d ı × ˆ ) + b d × ˆ ) = a d - b c . since ˆ ı ˆ , we have ˆ ı × ˆ ı = 0, ˆ × ˆ ı = +1, ˆ ı × ˆ = - 1, and ˆ × ˆ = 0. Note: sin 0 = 0 and sin 90 = 1. The magnitude oF k ˆ ı × ˆ k = (1)(1) sin 90 = 1 , and k ˆ × ˆ ı k = (1)(1) sin 90 = 1 . The vector product oF two vectors, when non-zero, is a vector. These two vector prod- ucts have direction given by ˆ ı × ˆ = ˆ k and ˆ × ˆ ı = - ˆ k , where ˆ k is a unit vector pointing along the positive z -axis. Thus the vectors ˆ ı × ˆ and ˆ × ˆ ı point in opposite directions. The result is ~ M × ~ N = ( a d - b c ) ˆ k .
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Pickering, Tracey – Homework 24 – Due: Mar 31 2006, noon – Inst: Drummond 2 003 (part 3 of 3) 10 points What is the direction of the vector product ~ M × ~ N ? 1.
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This note was uploaded on 06/23/2008 for the course PHY 303K taught by Professor Turner during the Fall '08 term at University of Texas at Austin.

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HW24 - Pickering, Tracey Homework 24 Due: Mar 31 2006, noon...

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