HW 08-solutions - gonzales (pag757) – HW 08 – Odell –...

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Unformatted text preview: gonzales (pag757) – HW 08 – Odell – (57000) 1 This print-out should have 25 questions. Multiple-choice questions may continue on the next column or page – find all choices before answering. 001 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 −−→ PD in terms of u and v . 1. −−→ PD = 2 v 2. −−→ PD = 2( u + v ) 3. −−→ PD = 2( u − v ) 4. −−→ PD = 2 v − u 5. −−→ PD = 2 u correct 6. −−→ PD = u + 2 v Explanation: By the parallelogram law for the addition of vectors we see that −−→ PD = 2 u . keywords: vectors, linear combination, vector sum displacement vector, parallelogram 002 10.0 points Determine a so that the vector u = (− 6 , 5 ) is a linear combination u = a v + b w of vectors v = (− 1 , 2 ) , w = (− 3 , − 1 ) . 1. a = − 1 2. a = − 3 3. a = 3 correct 4. a = 1 5. a = 2 Explanation: Since addition and scalar multiplication of vectors is carried out componentwise, we see that u = (− 6 , 5 ) = a (− 1 , 2 ) + b (− 3 , − 1 ) = (− a − 3 b, 2 a − b ) . Thus − a − 3 b = − 6 , 2 a − b = 5 . Consequently, after solving these for a we see that a = 3 . keywords: vectors, vector sum, linear combi- nation 003 10.0 points The parallelopiped in 3-space shown in Q P S C B A D R gonzales (pag757) – HW 08 – Odell – (57000) 2 is determined by its vertices P ( − 2 , − 4 , − 1) , Q (3 , − 4 , 1) , R (0 , , 2) , S ( − 3 , − 1 , 0) . Find the vector v represented by the directed line segment −−→ PC . 1. v = ( 4 , 3 , 3 ) 2. v = ( 6 , 7 , 6 ) 3. v = ( 7 , 8 , 6 ) 4. v = ( 1 , 7 , 4 ) correct 5. v = ( 7 , 4 , 5 ) 6. v = ( , 2 , 1 ) Explanation: As a vector sum, −−→ PC = −→ PR + −→ PS . But −→ PR = ( 2 , 4 , 3 ) , −→ PS = (− 1 , 3 , 1 ) . Consequently, v = −−→ PC = ( 1 , 7 , 4 ) . keywords: parallelopiped, 3-space, coordi- nates, vertex, directed line segment, vector sum 004 10.0 points For the vectors u and v shown in u v which of the following is equivalent to u − v ? 1. 2. 3. correct 4. 5. gonzales (pag757) – HW 08 – Odell – (57000) 3 6. Explanation: Vectors are equivalent when they have the same magnitude and direction. So we are looking for a vector having the same mag- nitude and direction as u − v . For this we write u − v = u + ( − v ) and use the parallelogram law for adding vec- tors. Note first that − v has the same length as v , but the opposite direction. Thus a vector equivalent to − v is indicated by the dashed arrow in u and so a vector equivalent to u − v is given by u In turn this is equivalent to u − v keywords: 005 10.0 points Determine the length of the vector − 2 a + b when a = (− 2 , 1 , − 1 ) , b = (− 1 , − 2 , − 1 ) ....
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This note was uploaded on 10/28/2009 for the course M 57000 taught by Professor Odell during the Fall '09 term at University of Texas at Austin.

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HW 08-solutions - gonzales (pag757) – HW 08 – Odell –...

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