# Math 221: Chapter 11 Exam Study Guide

- Given three vectors, find the following:
- A linear combination of vectors
- The norm of a vector
- The dot product
- The cross product
- The triple scalar product
- The projection of one vector onto another
- The angle between the vectors
- The direction cosines
- The direction angles for a vector

- Determine whether the two vectors are orthogonal, parallel, or neither.
- You are given the norms of the vectors, the different dot and cross products, and a triple scalar product, 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. Find the following:
- Find a cross product.
- Find a cross product.
- Find a triple scalar product (different order than what is given).
- Find the norm of a scalar product.
- A projection.
- Find the area of a parallelogram 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.

- 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 coordinates.
- Convert the rectangular equation into spherical coordinates.
- Convert the cylindrical equation into rectangular coordintates.
- Convert the spherical equation into rectangular coordinates.
- Identify the three-dimensional surface. Be precise in your identification. For example, a circular paraboloid (or cone) vs an eliptic paraboloid (or cone). 12 parts.
- Sketch the graph of the equation. There are 3 rectangular equations, 3 cylindrical equations, and 3 spherical equations.

## Notes

- There are 104 possible points on the exam, even though the exam is only worth 100 points.
- When angles are asked for, give them in degrees with at least one decimal place.

## Points per problem

# |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
10 |
11 |
12 |
Total |

Pts |
20 |
4 |
26 |
4 |
4 |
4 |
3 |
3 |
3 |
3 |
12 |
18 |
104 |