This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 yaxes. Both are parallel to the xyplane. c. False; lines can be skew in 3 dimensions. d. False; think about the x and yaxes again. Both are orthogonal to the zaxis. e. True. f. False; think about the xy, xz, and yzplanes. 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 ?...
View
Full
Document
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.
 Fall '08
 Keeran
 Calculus

Click to edit the document details