1. b Determine - X - and - X - for each pair of vectors. a. -= [3, -2, 9], - = [1, 6, 6] b. = [4.3, 5.7, -0.2], - = [12.3, -4.9, 8.8] 2. Use the cross product to determine the angle between the vectors - [4, 5, 2] and -> = [-2, 6, 1]. Verify using the dot product. 3 Find the volume of each parallelepiped, defined by the vectors -, ->, and -. a. - = [1, 4, 3], - = [2, 5, 6], and = = [1, 2, 7] b. - = [1, 1, 9], = = [0, 0, 4], and = = [-2, 0, 5] 4. Given - = [2, 2, 3], - = [1, 3, 4], and - = [6, 2, 1], evaluate each expression. a. - x -0-x- b . - X -0-x- u 5. When a wrench is rotated, the magnitude of the torque is 10 Nm. An 80 N force is applied 20 cm from the fulcrum. At what angle to the wrench is the force applied? 1. For the given points, a. vector equation of the line b. parametric equations of the line symmetric equation of the line d. scalar form of the line i. (1, 7) and (4, 10) ii. (-3, 5) and (-2, -8) 2 . Write the parametric and symmetric, and scalar equations for each vector equation. a . [x, y] = [1, -3] + t[2, 5] b. [x, y] = [8, 3] + t[-5, 2] You are given the vector form of the line [x, y] = [3, 1] +t[-2, 5] and a Point P. a w When you write the parametric equations of the line, what should you notice about the value of t? Are the following points on the line, i. (-1, 11) ii. (9, -15) Write the parametric and symmetric equations for each vector equation. a . [x, y, z] = [3, 0, -1] + t[6, -9, 1] b. [x, y, z] = [11, 2, 0] + t[3, 8, 2] 5 . Consider a plane with direction vectors [9, 4, 6] and [-8, 2, -7]. The plan passe through the point (10, 1, 5). Find the vector form and the parametric form of the equation of the plane. 1. Solve the following linear systems. a. [x, y] = [1, 6] + s[3, -2] b. [x, y] = [-12, -7] + s[8, -5] [x, y] = [4, 4] + t[-6, 4] [x, y] = [2, -1] + t[3, -2] C. [x, y] = [16, 1] + s[5, 1] d. [x, y] = [11, 12] + s[2, 7] [x, y] = [-7, 12] + t[-7, 3] [x, y] = [2, 3] + t[1, 4] 2 . Find the intersection of the line [x, y, z] = [2, 1, 0] + s[-1, 1, 3] and the plane 3x - 2y + z = 10. 3. Find the intersection of the line [x, y, z] = [1, -2, -1] + s[2, 3, 4] and the plane x + 2y - 2z = 5. 4. Write the equations of a line and a plane that intersect at the point (3, -8, 6)