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Please do these problems on separate sheets of paper and staple them together. Section 11.1 1. Plot the points (1, 0, 3), (0, −2, −1), (−1, 5, 2) on the same 3-dimensional coordinate system. 2. Find the distance between the points (−10, 3, 4) and (−2, 1, 5). 3. Find the center and radius of a sphere with equation x2 +y 2 +z 2 +2x−6y −10z +34 = 0 4. Sketch the graph of 3x − 4y + 2z = 24. Make sure to label the axis intercepts. 5. Describe the graph of the equation xyz = 0. 6. Find an equation whose solutions are the set of all points that are the same distance from the points (1, 2, −3) and (2, 3, 2). 7. Sketch the graph corresponding to the parametric equations x = 2 + t, y = 3 − t, z = 6 − 2t for −1 ≤ t ≤ 2 8. Find the length of the arc traced out by the parametric equations x = 2 cos(t), y = 2 sin(t), z = 3t for −π ≤ t ≤ π. (Hint: The arc length formula is Z bp (f 0 (t))2 + (g 0 (t))2 + (h0 (t))2 dt L= a where f , g, and h are the functions of t corresponding to x, y, and z.) Section 11.2 1. Draw each pair of vectors in a 2-dimensional coordinate plane, then do the indicated operation both algebraically and using your drawing. (a) u =< 4, 2 >, v =< 1, −3 >, calculate u + v (b) u =< 5, 0 >, v =< 4, −2 >, calculate u − 2v (c) u =< 2, 6 >, v =< −3, 4 >, calculate ||u|| and ||v|| 2. Find a unit vector in the direction of the vector < 3, 7, 5 >. 3. A 200 pound weight rests on a frictionless 30 degree incline. Someone standing on the incline above the weight’s position holds the weight in place with a rope and pulls parallel to the incline. With how much force must they pull to keep the weight in place? 1 4. The vectors u =< 3, 2 > and v =< −1, 5 > form a different basis for the plane than the standard basis with i and j. This means that any vector in 2 dimensions may be written as a linear combination of u and v. Write the vector < 5, 3 > as a combination of these two vectors. Section 11.3 1. For a =< 3, 1, 0 > and b =< −2, 4, −1 >, calculate the following: (a) 2a − 4b (b) a · b (c) ||a||2 − a · a (d) Then angle between a and b (e) The projection of a onto b, projb a (f) The projection of b onto a, proja b (g) The equation of the plane going through (5, 0, −3) and perpendicular to b. (h) An example of a vector that is perpendicular to both a and b. 2. Do Problem 1 again but for the vectors a =< 1, −1, 4 > and b =< 2, 6, 1 >. 3. For each expression, either calculate it or explain why it does not make sense. Use p =< −1, 3 >and q =< 5, 1 >. (a) (p · q) + q (b) (p + q) · p (c) 2 + p · p (d) (q · p)||q|| (e) (q − 2p)2 2