# Math 221: Chapter 11 Exam Study Guide

1. Given three vectors, find the following:
1. A linear combination of vectors
2. The norm of a vector
3. The dot product
4. The cross product
5. The triple scalar product
6. The projection of one vector onto another
7. The angle between the vectors
8. The direction cosines
9. The direction angles for a vector
2. Determine whether the two vectors are orthogonal, parallel, or neither.
3. 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:
1. Find a cross product.
2. Find a cross product.
3. Find a triple scalar product (different order than what is given).
4. Find the norm of a scalar product.
5. A projection.
6. Find the area of a parallelogram formed from two vectors.
7. Find the volume of the parallelpiped formed from three vectors.
8. Find the magnitude of a projection.
9. Find the angle between two vectors.
10. Find the parametric equations of a line passing through the given point in the direction of a vector.
11. Find the equation of a plane containing a point and two vectors.
12. Find the distance between a point and a plane.
13. Find the distance between a point and a line.
4. Find parametric and symmetric equations for the line passing through the two given points.
5. Find the equation of a plane passing through the three given points.
6. Find the distance between a point and a plane.
7. Convert the rectangular equation into cylindrical coordinates.
8. Convert the rectangular equation into spherical coordinates.
9. Convert the cylindrical equation into rectangular coordintates.
10. Convert the spherical equation into rectangular coordinates.
11. 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.
12. 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

 # Pts 1 2 3 4 5 6 7 8 9 10 11 12 Total 20 4 26 4 4 4 3 3 3 3 12 18 104