# Math 221: Chapter 11 Exam Study Guide

- Find the component form of a vector with the given magnitude and angle from the positive x-axis.
- Find the equation of the sphere with the given center and radius.
- Find a unit vector in the direction of the given vector.
- Given three vectors, find the following: linear combination of vectors, norm of a vector, dot product, cross product, triple scalar product, projection of one vector onto another, the angle between the vectors, the direction cosines for a vector.
- Determine whether the two vectors are orthogonal, parallel, or neither.
- Find parametric and symmetric equations for the line passing through the two given points.
- Find the equation of a plane passing through the three given points.
- Find the distance between a point and a plane.
- Convert the rectangular equation into cylindrical and spherical coordinates.
- Convert the cylindrical and spherical equations into rectangular coordinates.
- Find a cross product.
- Find a cross product.
- Find a triple scalar product.
- Find the norm of a scalar product.
- Find the area of a parallelgram formed from two vectors.
- Find the volume of the parallelpiped formed from three vectors.
- Find the magnitude of a projection.
- Find the angle between two vectors.
- Find the parametric equations of a line passing through the given point in the direction of a vector.
- Find the equation of a plane containing a point and two vectors.
- Find the distance between a point and a plane.
- Find the distance between a point and a line.
- Identify the three-dimensional surface. Be precise in your identification. For example, a circular paraboloid (or cone) vs an eliptical paraboloid (or cone). 8 parts.
- Sketch the graph of the equation. There are 3 rectangular equations, 3 cylindrical equations, and 3 spherical equations.

## Notes

- For questions 11-22, you are given the norms of the vectors, the different dot and cross products, but not the vectors themselves (except for one of the three vectors). Where it asks for a cross product, it will be a different cross product than what is given, so you will need to know the rules for dot and cross products.
- For question 24, you do not need to convert it into rectangular coordinates, but you may if it helps you graph it.

## Points per problem

# |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
10 |
12 |
13 |
14 |
15 |
16 |
17 |
18 |
19 |
20 |
21 |
22 |
23 |
24 |
Total |

Pts |
3 |
3 |
3 |
16 |
4 |
4 |
3 |
3 |
4 |
4 |
3 |
3 |
3 |
3 |
3 |
3 |
3 |
3 |
3 |
3 |
3 |
8 |
9 |
100 |