Math 242, Spring 2019, Assignment 2

No doubt many people feel that the inclusion of mathematics among the arts is unwarranted. The strongest objection is that mathematics has no emotional import. Of course this argument discounts the feelings of dislike and revulsion that mathematics induces....

- Morris Kline, Mathematics in Western Culture

Read:

1. Section 12.3.
2. Section 12.4.

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

1. Scalar multiple (of a vector).
2. Zero vector.
3. Standard unit vectors (i.e. $\vec{i}, \vec{j},$ and $\vec{k}$).
4. Dot product (of two vectors).
5. Direction angles (of a vector).
6. Direction cosines (of a vector).
7. $\mathrm{proj}_{\vec a}(\vec b)$ (the vector projection of $\vec b$ along $\vec a$).
8. $\mathrm{comp}_{\vec a}(\vec b)$ (the scalar projection of $\vec b$ along $\vec a$).
9. $\vec{a}\times\vec{b}$ (the cross product of $\vec a$ and $\vec b$).

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

1. Properties of vectors (in a box on page 802).
2. Properties of the dot product (in a box on page 807).
3. Geometric interpretation of the dot product (theorem 3, page 808).
4. Formula for the angle between two vectors.
5. Formulas for the direction cosines of a vector.
6. Formula for the vector projection.
7. Formula for the scalar projection.
8. Theorem concerning orthogonality of $\vec{a}\times\vec{b}$ to $\vec a$ and $\vec b$.
9. Right hand rule.
10. Geometric interpretation of the cross product (theorem 9 on page 817).
11. Criterion for two non-zero vectors to be perpendicular, in terms of the dot product.
12. Criterion for two non-zero vectors to be parallel, in terms of the cross product.
13. Properties of the cross product (in a box on page 819).

Solve the following problems:

1. Section 12.3, problems 3, 5, 7, 9, 15, 17, 33, 39, and 41.
2. Section 12.4, problems 1, 3, 9, 11, 15, and 19.

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