2313-practice-mid1-soln

2313-practice-mid1-soln - MAC 2313 Fall 2010 — Midterm 1...

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Unformatted text preview: MAC 2313, Fall 2010 — Midterm 1 Review Problems Solutions - 1 1 Explain why the following statements are true/false in 3 dimensions. a. Two planes either intersect or are parallel. b. Two lines parallel to a plane are parallel. c. Two lines either intersect or are parallel. d. Two lines orthogonal to a third line are parallel. e. A plane and a line either intersect or are parallel. f. Two planes orthogonal to a third plane are parallel. Solution: a. True. b. False; think about the x- and y-axes. Both are parallel to the xy-plane. c. False; lines can be skew in 3 dimensions. d. False; think about the x- and y-axes again. Both are orthogonal to the z-axis. e. True. f. False; think about the xy-, xz-, and yz-planes. 2 Is the angle between the vectors a = ( 3 , − 1 , 2 ) and b = ( 2 , 2 , 4 ) acute, obtuse, or right? Solution: Since a · b = 6 − 2 + 8 = 12 > 0 and a · b = | a || b | cos θ , we see that cos θ > 0, so the angle between a and b is acute. 3 Find the area of the parallelogram whose vertices are ( − 1 , 2 , 0), (0 , 4 , 2), (2 , 1 , − 2), and (3 , 3 , 0). Solution: Label the points P , Q , R , and S . Then −−→ PQ = ( 1 , 2 , 2 ) , −→ PR = ( 3 , − 1 , − 2 ) and −→ PS = ( 4 , 1 , ) . It follows that the parallelogram is determined by −−→ PQ and −→ PR , so it’s area is | −−→ PQ × −→ PR | = |(− 2 , 8 , − 7 )| = √ 4 + 64 + 49 = √ 117. MAC 2313, Fall 2010 — Midterm 1 Review Problems Solutions - 2 4 If a and b are both nonzero vectors and a · b = | a × b | , what can you say about the relationship between a and b ? Solution: We are given that a · b = | a × b | , and we know that a · b = | a || b | cos θ , while | a × b | = | a || b | sin θ It follows that we must have cos θ = sin θ , and the only value of θ which satisfies this is θ = π/ 4, so the two vectors are at a 45 ◦ angle to each other. 5 Suppose proj b a = 3 i − 4 j and that the angle between a and b is obtuse. What is comp b a ?...
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This note was uploaded on 12/10/2011 for the course MAC 2313 taught by Professor Keeran during the Fall '08 term at University of Florida.

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2313-practice-mid1-soln - MAC 2313 Fall 2010 — Midterm 1...

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