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.