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Midterm 1: Practice Exam Math 2210-006 Spring 2016 Warnings: (1) Please bring your student ID with you during midterm exams. (2) No references or calculators can be used for midterm exams. Practice Exam Z 3 xydx along the curve x = t − 1, y = t + 1. (1) Calculate 1 (2) Find the arc length of this curve: x = t3/2 , y = t3/2 , z = t; 2 ≤ t ≤ 4 (3) Two vectors are given: u =< 1, −3, 4 > and v =< −2, 1, −1 > (a) Calculate the angle between the two vectors; (b) The projection of u onto v and that of v onto u; (c) Suppose u is orthogonal to v +cw, where w =< 0, 0, 1 >. Determine the scalar c. (4) Find the plane through (2, 4, 3) and is parallel to: (a) xy plane; (b) the plane x − y + 6z = 213 (5) Find the distance: (a) from the point (1, 0, −3) to the plane 3x − y + z = 4 (b) between two parallel planes x − 2y + z = 3 and x − 2y + z = 5 (6) Find the measure of the angle ABC, where the three points are A(0, 1), B(2, 3) and C(−1, 5). (7) Find the directional angles of ~x =< −1, 2, 5 > with respect to î, ĵ and k̂. (8) Find the vector that is orthogonal to both ~u =< 1, −1, 0 > and ~v =< 0, 0, 1 >. (9) For the sphere that uses (1, 2, 0) and (3, 2, 2) as endpoints of a certain diameter, find its equation.