1. Find the distance between (-3,2,5) and (6,1,2).
2. Find the area of the triangle with the following vertices (0,2,-1), (2,2,-3), and (-1,0,5).
3. Calculate the distance between the point (2,-1,4) and the plane 3x + 5y - 2z = 11.
4. Find an equation of the plane that passes through the points (2,-1,2), (-3,7,1), and (3,1,2).
5. Calculate the angle between u = 3i - j + 2k and v = -i + 4j + 5k.
6. Find the midpoint of the line segment connecting (2,1,5) and (4,-1,3).
7. Find the area of the parallelogram having u = i + 3j + 2k and v = 5i - 4j + k as adjacent sides.
8. Find an equation for the sphere that has points (2, 1, 7) and (4, -2, 1) as endpoints of a diameter.
9. A vector v has initial point (3,1,6) and terminal point (-2,1,-8). Write v in component form.
10. A vector v has initial point
(3,1,6) and terminal point (-2,1,-8). Write v as a linear
combination of
the standard
unit vectors.
11. A vector v has initial point (3,1,6) and terminal point (-2,1,-8). Find the magnitude of v.
12. A vector v has initial point
(3,1,6) and terminal point (-2,1,-8). Find the unit vector in the
direction
of v.
13. A vector v has initial point
(3,1,6) and terminal point (-2,1,-8). Find the unit vector in the
direction
opposite
that of v.
14. Find the center and radius of the
sphere
given by the equation
3x2
+ 3y2 + 3z2 - 12x - 15y - 21z - 100 = 0.
15. Find the initial point of the vector v = 4i - 2k if the terminal point is (1,5,2).
16. Let u = -i + 2j - 3k and w = 2i - j - k. Calculate a vector orthogonal to both u and v.
17. Find the volume of a parallelepiped
that has u = 2j + k, v = 2i + 4k,
and
w
= i + 2j - 6k as
adjacent
sides.
18. Find parametric equations for the
line
through the point (2,-3,1) and parallel to the line
x
- 2 = y + 5 = z - 1 .
3
4
2
19. Sketch the plane: 2x - 5y + 3z = 30