Problem 1. (a) Find the angle between the vectors (-1, 1,0) and (-6,3, -6). (b) Find t such that (7, 3,5) and (2, 1, t) are perpendicular. (c) Find & such that the angle between (t, 5, -1) and (1, -1,5) is acute (between - and ). Problem 2. Let P(a, 1, 2), Q(0, 2, 1), R(-a, -1, 4) be points in 3D. (a) For what value(s) of a is the angle PQR a right angle? (b) For the value(s) of a obtained in part (a), find the area of the triangle PQR. Problem 3. If a = (4,0, -1), find a vector b such that (proj,b) = 2. Problem 4. Find the area of the triangle in 3D with vertices P(3,0, -1), Q(5, 1,0), and R(-1, 1, -1) in two ways: one using the dot product and one using the cross product. Problem 5. (a) Find a vector v such that (-1,0, 1) x v = (0,6, -2) or explain why there is no such vector. (b) Find a vector v such that (0, 1, 3) x v = (0,6, -2) or explain why there is no such vector. Problem 6. Consider A(1, 0,5), B(3, 1, 3), C(2, -3, 4), D(0, -1, 3) points in 3D. Find the volume of the parallelepiped with adjacent edges AB, AC, and AD. Problem 7. Consider the points A(-1, 0, 1), B(0, 1, -4), C(1, 1, -1), D(2, 2, -6) in 3D. Do A, B, C, and D lie in the same plane in 3D? Problem 8. Let a and b be 3D vectors such that a . b = v3 and a x b = (1, 2, 2). Find the angle between a and b