MATH 2110 – EXAM 1 REVIEW PROBLEMS
In addition to these problems, be sure to review your notes and homework when
preparing for the exam.
1. Find the distance between the points (2, -1, 3) and (1, 0, -2).
2. Given vectors u = <1, 2, 3> and v = <2, 2, -1>, find
a.) u + v b.) u – v c.) 3u + 2v
d.) u • v e) the angle between u and v
3. Find the unit vector that has the same direction as i – 4j + 8k
4. a.) Find the equation of the sphere that has the points (5, -2, 3) and (0, 4, -3) as
endpoints of a diameter.
b.) Find center and radius of the sphere x
2
+ y
2
+ z
2
+ 9x -2y +10z = -19
5. Given the vertices P = (1, -3, -2), Q = (5, -1, 2), R = (-1, 1, 2) of a triangle,
a.) Determine whether the triangle is a right triangle, isosceles triangle, or neither
b.) Find the area of the triangle
c.) Find the equation of the plane containing P, Q, and R
6. Given the points P = (2, -1, -2) and Q = (3, 1, 4), find
a.) the parametric form and symmetric equations of the line containing P and Q
b.) a third point on the line
7. What is known about , the angle between two nonzero vectors a and b, if a • b < 0?
8. A toy wagon is pulled by exerting a force of 15 pounds on a handle that makes a 30°
angle with the horizontal. Find the work done in pulling the wagon 50 feet.
9. Determine if the following lines are parallel, intersecting, or skew. If they intersect,
find the point of intersection.
L
1
: x = 4t + 2, y = 3, z = -t + 1
L
2
: x = 2s + 2, y = 2s + 3, z = s + 1
10. Find the intersection of the line r = <1, 0, 2> + t<2, -2, 1> and the plane
3x + 4y + 6z = 7
11. Find an equation of the plane that passes through the point (4, 0, -2) and contains the
line x = 10 – 3t, y = 10 + 8t, z = 6 + 7t.
12. Find parametric equations for the line of intersection of the planes:
3x + 2y – z = 7
x – 4y + 2z = 0
