Merced College; Don Power

MULTIVARIATE CALCULUS, CH 12 - STUDY GUIDE

1. Find a vector that is normal to the plane 3x-5y+7z-2=0.

2. For the line x=4-2t, y=-1+7t, z=2-t

a. Find (a) a point on the line

b. Find a vector parallel to the line.

c. Write an equation of the plane containing the point (-1, 7,2) and the line
(you will need your results from parts a and b to complete part c)

3. For the line y = 3x+7 [in the xy-plane]:

a. Find vectors parallel to (and perpendicular to) the line.

b. Find the distance between the point (-1,2) and the line

4. a. Find the center and radius of the sphere x²+y²+z²+8x-4y+2z-7=0.

b. Write an equation of the sphere whose center is (3,-2,-5) that passes through (2,1,-1)

5. a. For the triangle with vertices at P(4, -2,1), Q(-4,3,8) and R(9, -3,1), find the cosine of the angle at R.

b. Find the angle.

c. Find the area of the triangle.

d. Find the equation of the plane containing these points

6. a. Find a unit vector that is orthogonal to both u=7i-2j-8k and v=-2i+1j-8k.

b. Determine whether u and v are parallel, orthogonal, or neither

7. a. Write the vector from (-4,1,3) to (5,6, -1)

b. Write or parametric equations of the line through these points.

8. Find an equation of the surface obtained by revolving 5y²+z²=3 about the y-axis. Sketch the resulting surface. Is it a paraboloid, an ellipsoid, a hyperboloid, or none of these?

9. a. At what point does the line x=3+2t, y=7t, z=-2-t intersect the xz-plane? Hint: Which coordinate is 0?

b. At what angle does this line intersect the xz-plane? (One technique: 90^o − v dot n....why?)

10. Find the angle of intersection of the planes 4x-2y+z = 3 and 3x+2y-8z = 11.

11. Let u = 4i-2j+7k and v = -2i+j+k. Find

a. x, where 3u+x = v

b.

c. A vector of length 5 in the direction of v (Hint: 5 times a unit vector in the direction of v).

d. Scalars a and b such that au+bv = -2i+j+28k.

e. The projection of u onto v, and the vector component of u orthogonal to v.

12. Let u = 2i-j+k, v = 3i+3j-k, w = i+4j-2k

a. Find the volume of the oblique box (parallelepiped) determined by u, v, and w

b. Determine whether u, v, and w lie in the same plane

13. Identify the trace in the xy-plane, the xz-plane, and the yz-plane; and identify the 3-dimensional surface, for

a. 3x²-2y²-z²-6x+6=0

b.

14. Find the point of intersection of the lines

x = 2-t, y = 3+2t, z = 6+t and x = 5-4t, y = 4t+2, z = 13-4t. [You can't use t as the parameter for both lines]

15. Convert points from

a. Rectangular coordinates to cylindrical or spherical coordinates

b. Cylindrical or spherical coordinates to rectangular coordinates

17. Convert an equation from

a. Cylindrical or spherical coordinates to rectangular coordinates

b. Rectangular coordinates to cylindrical or spherical coordinates

18. Identify surfaces in cylindrical or spherical coordinates

Return to:  Merced College; Don Power               Updated 02/01/06 by Don Power