Unformatted text preview: f the graph depend on the numbers a and b?
x 2 y 2 c xy. In particular, you
should determine the transitional values of c for which the surface changes from one type of
quadric surface to another. 2. Use a computer to investigate the family of surfaces z 3. Members of the family of surfaces given in spherical coordinates by the equation 1 0.2 sin m sin n have been suggested as models for tumors and have been called bumpy spheres and wrinkled
spheres. Use a computer to investigate this family of surfaces, assuming that m and n are
positive integers. What roles do the values of m and n play in the shape of the surface?  13 Review ■ CONCEPT CHECK ■ 1. What is the difference between a vector and a scalar? 12. How do you ﬁnd the angle between two intersecting planes? 2. How do you add two vectors geometrically? How do you add 13. Write a vector equation, parametric equations, and symmetric them algebraically?
3. If a is a vector and c is a scalar, how is c a related to a geo metrically? How do you ﬁnd c a algebraically?
4. How do you ﬁnd the vector from one point to another?
5. How do you ﬁnd the dot product a b of two vectors if you know their lengths and the angle between them? What if you
know their components?
6. How are dot products useful?
7. Write expressions for the scalar and vector projections of b onto a. Illustrate with diagrams.
8. How do you ﬁnd the cross product a b of two vectors if you
know their lengths and the angle between them? What if you
know their components? 9. How are cross products useful?
10. (a) How do you ﬁnd the area of the parallelogram determined by a and b?
(b) How do you ﬁnd the volume of the parallelepiped determined by a, b, and c?
11. How do you ﬁnd a vector perpendicular to a plane? equations for a line.
14. Write a vector equation and a scalar equation for a plane.
15. (a) How do you tell if two vectors are parallel? (b) How do you tell if two vectors are perpendicular?
(c) How do you tell if two planes are parallel?
16. (a) Describe a method for deter...
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 Fall '09
 hamrick
 Linear Algebra, Vectors, Vector Space, Dot Product, (pp 828837)

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