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MATH 151 Engineering Mathematics I Week In Review Fall, 2015, Problem Set 2 (1.2, 1.3) JoungDong Kim 1. Find a · b (a) |a| = 2, |b| = 3, the angle between a and b is (b) a = h−1, −2i, b = h2, 8i. (c) a = i − j, b = i + 2j 1 π . 3 2. Find the angle between the vectors. (Answer in cos−1 form.) (a) a = h6, 0i, b = h5, 3i (b) a = i + j, b = 2j 3. Find the values of x such that the given vectors are orthogonal. (a) hx, xi, h1, xi (b) xi − 2j, xi + 8j 2 4. Find the scalar and vector projections of b onto a. (a) a = h3, −1i, b = h2, 3i (b) a = 2i − 3j, b = i + 6j 3 5. Find the distance from the point (3, 7) to the line y = 2x. 6. A constant force with vector representation F = 10i + 18j moves an object along a straight line from the point (2, 3) to the point (4, 9). Find the work done if the distance is measured in meters and the magnitude of the force is measured in newtons. 4 7. Eliminate the parameter to find the Cartesian equation of the curve. And sketch the curve and indicate with an arrow the direction in which the curve is traced as the parameter increases. (a) x = 2t − 1, y = 2 − t, −3 ≤ t ≤ 3. (b) x = 2t − 1, y = t2 − 1. (c) x = 3 cos θ, y = 2 sin θ, 0 ≤ θ ≤ 2π. 5 8. Find a vector equation, parametric equations, and a Cartesian equation for the line passing through (1, 3) and (−2, 7). 6 9. Determine whether the lines L1 and L2 given by the vector equations are parallel, perpendicular, or neither. If the lines are not parallel, find their point of intersection. L1 : r(t) = (−4 + 2t)i + (5 + t)j L2 : r(t) = (2 + 3t)i + (4 − 6t)j 7 10. An object is moving in the xy-plane and its position after t seconds is r(t) = ht − 3, t2 − 2ti. (a) Find the position of the object at time t = 5. (b) At what time is the object at the point (1, 8)? (c) Does the object pass through the point (3, 20)? (d) Find an equation in x and y whose graph is the path of the object. 8