# Final Winter 2015 Solutions - Northwestern University Math...

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Northwestern University Math 230 Final exam March 16, 2015 Name . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . NetID . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Student ID . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Instructions: This exam has 13 problems and 16 pages. Please make sure that all pages are in- cluded. Read each problem carefully. Write legibly. Cross off anything you do not wish graded. Write in complete sentences. You may not use books, notes, calculators or phones. You have 120 minutes to complete this exam. DO NOT write outside of the box on each page. Good luck!
Problem 1 (14 points) Consider the three vectors a = h 1, 2, 0 i , b = h- 1, 1, 1 i , c = h 2, 1, - 1 i . (a) (6 points) Compute a × b and a · b .
(b) (2 points) Are the vectors a and b parallel or orthogonal or neither? Circle the correct answer.
(c) (4 points) Compute ( a × b ) · c .
(d) (2 points) Are the vectors a , b and c coplanar, i.e., they lie on the same plane? Circle the correct answer. Yes
Problem 2 (15 points) Consider the line L 1 given by x = 1 - 2 t , y = 2 + t , z = - t . (a) (8 points) Does the line L 1 intersect the line L 2 given by x = 3 + 3 t , y = 2 - t , z = 1 + 4 t ? If so, find the point of intersection.
(b) (7 points) Does the line L 1 intersect the plane x + y + z = 2? If so, find the point (or points) of intersection.
Problem 3 (14 points, 7 points each) For each of the following curves in polar coordinates r and θ , (i) write an equation for the curve in Cartesian coordinates x and y , and (ii) sketch the curve on the xy -plane. (a) r = sin θ .