Math 221: Chapter 11 Exam Study Guide

1. Find the component form of a vector with the given magnitude and angle from the positive x-axis.
2. Find the equation of the sphere with the given center and radius.
3. Find a unit vector in the direction of the given vector.
4. 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.
5. Determine whether the two vectors are orthogonal, parallel, or neither.
6. Find parametric and symmetric equations for the line passing through the two given points.
7. Find the equation of a plane passing through the three given points.
8. Find the distance between a point and a plane.
9. Convert the rectangular equation into cylindrical and spherical coordinates.
10. Convert the cylindrical and spherical equations into rectangular coordinates.
11. Find a cross product.
12. Find a cross product.
13. Find a triple scalar product.
14. Find the norm of a scalar product.
15. Find the area of a parallelgram formed from two vectors.
16. Find the volume of the parallelpiped formed from three vectors.
17. Find the magnitude of a projection.
18. Find the angle between two vectors.
19. Find the parametric equations of a line passing through the given point in the direction of a vector.
20. Find the equation of a plane containing a point and two vectors.
21. Find the distance between a point and a plane.
22. Find the distance between a point and a line.
23. 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.
24. 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

 # Pts 1 2 3 4 5 6 7 8 9 10 12 13 14 15 16 17 18 19 20 21 22 23 24 Total 3 3 3 16 4 4 3 3 4 4 3 3 3 3 3 3 3 3 3 3 3 8 9 100