1) (10 points) Let point P be (2,3,-2) and OP be a position vector for point P. a) (1 point) Draw the 3-D coordinate axes and plot the point P b) (1 point) Draw the position vector OP for point P (1,1,2) on the plot drawn above. c) (2 points) Let a = -i + 2j + 3k and b = i - 2j + 3k. Find a . b d) (2 points) Find the angle between vectors a and b f) (2 pts) Find the scalar projection of b onto a g) (2 pts) Find the vector projection of b onto a10) (10 points) Let f(x,y,z) = xy? - 2x?yz2. a) (5 points) Find the gradient of f. b) (5 points) Find the maximum rate of change of fat (1,2,-1) Extra Credit (10 points) Find the minimum value of f(x,y) = xy given that x?+y?=5. 2) (10 points) Let a = -i - 2j + 3k, b = i - 2j + 3k, c = -i+ 2j - 3k. a) (5 points) Find a vector that is perpendicular to b and & b) (5 points) Find | a . (b x c) |. State its geometric interpretation. 3) (10 points) Find a vector equation (4 pts), parametric equation (3 pts), and symmetric equation (3 pts) for the line passing through (2,-1,1) that is parallel to the vector a = -i- 2j + 4k.4) (10 points) Find the equation of the plane a) that passes through (1,2,4) and is perpendicular to i - 2j +3k b) Find the equation of the plane that passes through the points (-3, 1, -2), (2,1,-1), and (1,-2,-3).5) (10 points) Find the following: a) (3 points) Find the tangent vector for the position vector r(t) = In(1-t)i + (t t3)j+e 3tk b) (3 points) Find the tangent vector at the point where t = 1. c) (2 points) Find the unit tangent vector at the point where t = 1. d) (2 points) Show that the position vector and tangent vector are perpendicular. 6) (10 points) Solve the following: a) (5 points) Evaluate the integral of So (Ci -e-2tt3j + sin(2t)k ) dt b) (5 points) Find the length of the curve for r(t) = ti + 2t?j +- k for 0 . Find the directional derivative Duf (x, y, z). 9) (10 points) Use implicit differentiation to find z and "2 for x6 + 2y2+ 324 = 1