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HW 12.4-solutions

# HW 12.4-solutions - morrell(mm59638 HW 12.4 radin(56025...

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morrell (mm59638) – HW 12.4 – radin – (56025) 1 This print-out should have 13 questions. Multiple-choice questions may continue on the next column or page – fnd all choices beFore answering. 001 10.0 points ±ind the value oF the determinant D = v v v v v v v 1 2 x 2 1 y 3 3 z v v v v v v v . 1. D = 3 x + 3 y + 3 z 2. D = 3 x 3 y 3 z correct 3. D = 3 x 3 y 3 z 4. D = 3 x + 3 y + 3 z 5. D = 3 x + 3 y 3 z 6. D = 3 x 3 y + 3 z Explanation: ±or any 3 × 3 determinant v v v v v v v A B C a 1 b 1 c 1 a 2 b 2 c 2 v v v v v v v = A v v v v v b 1 c 1 b 2 c 2 v v v v v B v v v v v a 1 c 1 a 2 c 2 v v v v v + C v v v v v a 1 b 1 a 2 b 2 v v v v v . Thus D = v v v v v v v 1 2 x 2 1 y 3 3 z v v v v v v v = v v v v v 1 y 3 z v v v v v + 2 v v v v v 2 y 3 z v v v v v + x v v v v v 2 1 3 3 v v v v v = (1 z + 3 y ) + 2 ( 2 z 3 y ) + x (3) . Consequently, D = 3 x 3 y 3 z . keywords: determinant 002 10.0 points ±ind the cross product oF the vectors a = 2 i + 3 j + k , b = 3 i + 3 j + 3 k . 1. a × b = 6 i + 10 j + 3 k 2. a × b = 6 i + 9 j + 2 k 3. a × b = 7 i + 3 j + 3 k 4. a × b = 6 i + 3 j + 3 k correct 5. a × b = 7 i + 3 j + 2 k 6. a × b = 7 i + 10 j + 2 k Explanation: One way oF computing the cross product ( 2 i + 3 j + k ) × ( 3 i + 3 j + 3 k ) is to use the Fact that i × j = k , j × k = i , k × i = j , while i × i = 0 , j × j = 0 , k × k = 0 . ±or then a × b = 6 i + 3 j + 3 k . Alternatively, we can use the defnition a × b = v v v v v v i j k 2 3 1 3 3 3 v v v v v v = v v v v 3 1 3 3 v v v v i v v v v 2 1 3 3 v v v v j + v v v v 2 3 3 3 v v v v k

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morrell (mm59638) – HW 12.4 – radin – (56025) 2 to determine a × b . 003 10.0 points Determine all unit vectors v orthogonal to a = 3 i + 4 j + k , b = 9 i + 10 j + 2 k . 1. v = ± p 2 7 i + 3 7 j 6 7 k P correct 2. v = 2 i 6 j + 3 k 3. v = ± p 2 7 i 6 7 j + 3 7 k P 4. v = 2 7 i 6 7 j + 3 7 k 5. v = 2 7 i 3 7 j + 6 7 k 6. v = 2 i 3 j + 6 k Explanation: The non-zero vectors orthogonal to a and b are all of the form v = λ ( a × b ) , λ n = 0 , with λ a scalar. The only unit vectors orthog- onal to a , b are thus v = ± a × b | a × b | . But for the given vectors a and b , a × b = v v v v v v i j k 3 4 1 9 10 2 v v v v v v = v v v v 4 1 10 2 v v v v i v v v v 3 1 9 2 v v v v j + v v v v 3 4 9 10 v v v v k = 2 i 3 j + 6 k .
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