1. Let A, B, and C be points in the plane 3x - 5y + 6z = ! ! ! 7, and a , b and c be the corresponding vectors (based ! ! ! j at the origin). Compute ( a + 4 b 7c ) (3i 5 + 6 k ) . 2. Consider the points A=(1,0,1), B=(0,2,2) and C=(0,-1,0). Let L1 be the line through A and B, and let L2 be the line through A and . Find the angle between L1 and L2. C 3. A moving particle travels through space with velocity at ! time t given by v (t) = <3et, 4e-t, 1> and passes through the point (4,3,3) at time t = 0. (a) Find the particle's speed at time t = 0. ! (b) Find the function p(t) that gives the particle's position at time t. ! (c) Find the particle's acceleration a(t) . ! ! n 4. Suppose f , g : are both differentiable at t. Using the definition of the dot product and our component-wise differentiation theorem, prove the product rule for the dot product: ! ! d ! ! ! ! f (t) g(t) = f '(t) g(t) + f (t) g'(t). dt ( ) Remember to justify each step briefly! x2 + y 2 = 1 in 2 is shown below. Suppose 5. The ellipse 9 ! f : [0, 2] 2 is a smooth parameterization of the ellipse that traces out counterclockwise starting at (3,0). (a) Sketch the unit tangent T and the principal normal N at the point A on the graph. Label them clearly. (b) Sketch the osculating circle at the point C on the graph. (c) List the points A, B, C in order of increasing curvature (t). Explain your answer briefly in words