Math 221-2 First Midquarter Exam Name_________________ Treibergs Wednesday, October 11, 1995 This is a closed book exam. No books, papers or calculators. 1.___________/20 [20] indicates points per problem. 2.___________/20 3.___________/20 1. [20] Find the distance between the line 4.___________/20 x = (1, 0, 1) + s (1, 3, 3) 5.___________/20 and the point (5, 4, 3). Total___________/100 2.[20] Suppose x and y are perpendicular vectors in R3. Prove the Pythagorean Theorem: | x |2 + | y |2 = | x + y |2 3. Determine whether each of the following systems of equations have solutions. If there are solutions, find them. If the set of solutions forms a line or plane then represent it parametrically. a.[10] x + 2y + 3z = 4 2x + y + 2z = 9 3x + 9y + 13z = 11 b.[10] x + 2y + 3z = 4 2x + y + 2z = 9 3x + 9y + 13z = 0 4.[20] Find the point of intersection in R3 between the plane x = (1, 2, 3) + r (1, 1, 1) + s (2, 1, 1) and the line x = (4, 4, 4) + t (1, Ð1, 1). 5.[20] Find an equation of the form ax + by + cz = d for the plane passing through the line x = (1, 1, 1 ) + s (1, 3, 2) and through the point (6, 5, 4).