Math 242, Spring 2019, Assignment 3

We admit, in geometry, not only infinite magnitudes, that is to say, magnitudes greater than any assignable magnitude, but infinite magnitudes infinitely greater, the one than the other. This astonishes our dimension of brains, which is only about six inches long, five broad, and six in depth, in the largest heads.

- Voltaire

Read:

1. Section 12.5.
2. Section 12.6.

Carefully define the following terms, then give one example and one non-example of each:

1. Scalar triple product (of three vectors $\vec a, \vec b,$ and $\vec c$).
2. Cylinder.
3. Quadric surface.

Carefully state the following theorems (you need not prove them):

1. Theorem relating the magnitude of the cross product to the area of a parallelogram (on page 818).
2. Theorem relating scalar triple products to volumes of parallelepipeds (on page 820).
3. Parametric equations for a line through a given point, with a given direction vector (on page 824).
4. Symmetric equations for a line though a given point, with a given direction vector (on page 825).
5. Formula parametrizing the line segment between two points (on page 826).
6. Vector equation for a plane through a given point, with a given normal vector.
7. Scalar equation for a plane through a given point, with a given normal vector.

Solve the following problems:

1. Section 12.4, problems 27, 33, 35, and 37.
2. Section 12.5, problems 3, 5, 7, 23, 25, 27, and 31.
3. Section 12.6, problems 1, 3, 5, 11, 13, 17, 21, 23, 25, 27, 31, 33, 35, and 37.

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