# Math 221: Study Guide - Chapter 12

1. Find parametric equations for the lines passing through the given points.
2. Find the angle between two vectors.
3. Find the equation of the plane passing through the given point with the given normal vector.
4. Find the point of intersection of the line and the plane.
5. Find the distance between the point and the plane.
6. Find the equation of the plane containing the given three points.
7. Convert the equation from cylindrical coordinates into rectangular coordinates.
8. Convert the equation from spherical coordinates into rectangular coordinates.
9. Convert the equation from rectangular coordinates into both cylindrical and spherical coordinates.
10. Find the volume of the parallelpiped with the given adjacent sides.
11. Identify whether the trace of the surface in the given plane is a circle, ellipse, parabola, or hyperbola. Three parts.
12. Given two vectors, find the sum, difference, scalar product, norm, dot product, cross product, unit vector, and parallel vector having a particular length.
13. Identify, but do not sketch, the three dimensional surface. Most of these are quadrics, but you could also have cylinders, points, no graphs, intersecting lines, and other degenerate cases. Be specific, that is, if it is a circular paraboloid as opposed to an elliptical paraboloid. Ten parts.
14. Given three vectors, simplify the expressions. These could involve projections, dot products, cross products, and scalar triple products. Three parts.
15. Given some dot products and cross products, simplify the expressions. These could include norms, norms of scalar products, scalar triple products, norms of projections, angles between vectors, and areas of parallelograms. The actual vectors are not given. Nine parts.

## Notes

• You may use the technology exercise on Quadric surfaces that you prepare for this chapter on the test.

## Points per problem

 # Pts 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Total 3 4 4 4 4 6 4 4 8 4 6 14 10 9 18 100