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M408D Quest Homework 7-solutions

# M408D Quest Homework 7-solutions - hernandez(ah29758 –...

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Unformatted text preview: hernandez (ah29758) – M408D Quest Homework 7 – pascaleff – (54550) 1 This print-out should have 19 questions. Multiple-choice questions may continue on the next column or page – find all choices before answering. 001 10.0 points Find a unit vector n with the same direction as the vector v = 2 i + 3 j + 6 k . 1. n = 2 7 i − 3 7 j − 6 7 k 2. n = 2 7 i + 3 7 j + 6 7 k correct 3. n = 1 5 i + 3 10 j + 3 5 k 4. n = 2 9 i − 1 3 j − 2 3 k 5. n = 2 9 i + 1 3 j + 2 3 k 6. n = 1 5 i − 3 10 j − 3 5 k Explanation: The vector v = 2 i + 3 j + 6 k . has length | v | = radicalbig 2 2 + 3 2 + 6 2 = √ 49 = 7 . Consequently, n = v | v | = 2 7 i + 3 7 j + 6 7 k is a unit vector having the same direction as v . 002 10.0 points Determine the length of the vector 2 a + b when a = ( 1 , 3 , − 2 ) , b = (− 3 , 1 , 3 ) . 1. length = 7 2. length = √ 51 correct 3. length = √ 47 4. length = √ 43 5. length = 3 √ 5 Explanation: The length, | c | , of the vector c = ( c 1 , c 2 , c 3 ) is defined by | c | = radicalBig c 2 1 + c 2 2 + c 2 3 . Consequently, when a = ( 1 , 3 , − 2 ) , b = (− 3 , 1 , 3 ) , and c = 2 a + b = (− 1 , 7 , − 1 ) , we see that | 2 a + b | = √ 51 . 003 10.0 points Find all scalars λ so that λ ( a − 2 b ) is a unit vector when a = ( 2 , 3 ) , b = (− 1 , 2 ) . 1. λ = − 1 17 2. λ = 1 √ 17 3. λ = − 1 √ 17 4. λ = ± 1 17 hernandez (ah29758) – M408D Quest Homework 7 – pascaleff – (54550) 2 5. λ = ± 1 √ 17 correct 6. λ = 1 17 Explanation: A vector c = ( c 1 , c 2 ) is said to be a unit vector when | c | = radicalBig c 2 1 + c 2 2 = 1 . But for the given vectors a and b , λ ( a − 2 b ) = λ ( 4 , − 1 ) = ( 4 λ, − λ ) . Thus | λ ( a − 2 b ) | = radicalBig λ 2 ((4) 2 + ( − 1) 2 ) = | λ | radicalBig (4) 2 + ( − 1) 2 = | λ | √ 17 . Consequently, λ ( a − 2 b ) will be a unit vector if and only if λ = ± 1 √ 17 . keywords: vector sum, length, linear combi- nation, unit vector, 004 10.0 points When u , v are the displacement vectors u = −−→ AB , v = −→ AP , determined by the parallelogram A B C D P Q R S O express −→ AS in terms of u and v . 1. −→ AS = 2 v 2. −→ AS = 2( u + v ) correct 3. −→ AS = u + 2 v 4. −→ AS = 2( u − v ) 5. −→ AS = 2 v − u 6. −→ AS = 2 u Explanation: By the parallelogram law for the addition of vectors we see that −→ AS = 2( u + v ) . keywords: vectors, linear combination, vector sum displacement vector, parallelogram 005 10.0 points Find the vector v having a representation by the directed line segment −−→ AB with respect to points A (1 , − 3 , − 1) , B ( − 4 , 2 , − 2) . 1. v = (− 5 , − 5 , − 1 ) 2. v = ( 5 , 5 , 1 ) 3. v = (− 3 , − 1 , − 3 ) 4. v = ( 3 , − 1 , 3 ) 5. v = (− 5 , 5 , − 1 ) correct 6. v = (− 3 , 1 , − 3 ) Explanation: Since −−→ AB = (− 4 − 1 , 2 + 3 , − 2 + 1 ) , hernandez (ah29758) – M408D Quest Homework 7 – pascaleff – (54550) 3 we see that...
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M408D Quest Homework 7-solutions - hernandez(ah29758 –...

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