Math 221-2 First Midquarter Exam Name Sample Treibergs Tue., Jan. 16, 1995 This is a closed book exam. No books, papers calculators. There are [100] total points. 1. [20] Find the parametric form x = x0 + su + tv of the plane passing through the line x = (1,2,3 ) + s(1,0,Ð1) and through the point (3, 2 ,3). 2.[20] Let u = (1,0,2), v = (3,2,0), w = (0,1,1). Write (17,11,13) as a linear combination of u, v, w. 3.[20] Determine whether the following equations are consistent or inconsistent. Find all possible solutions to the consistent systems x + 2y + 3z = 4 x + 2y + 3z = 4 2x + y + 2z = 1 2x + y + 2z = 1 3x + z = 3 3x + z = 2 x + 2y + 3z = 2 2x + 4y + 6z = 4 3x + 6y + 9z = 6 4.[20] Find the equation of the line through (1,2,3) and perpendicular to the plane passing through the points (1,1,1), (1,0,0), (1,1,0). 5.[20] Find the parametric form of the line of intersection between the planes x + 2y + 3z = 1 2x + 3y + 4z = 7 E1. Find the angle between the planes in Problem 5. E2. For all possible vectors u, v, w in R3 such that | u + v + w | > |u| + |v| + |v|. E3. Find the distance between the lines x = (1,2,3) + s(1,1,1) y = (3,2,1) + t(1,Ð1,Ð1) E4. Let u=(1,1) and v = (1,Ð1) in R2. Show that u and v are linearly independent. E5. Show that there are infinitely many solutions to this homogeneous system. Represent all solutions parametrically x - y + 3z = 0 2x + 3y + 4z = 0 7x + 8y + 15z = 0