Math 242, Fall 2013, Assignment 2

By one of those caprices of the mind, which we are perhaps most subject to in early youth, I at once gave up my former occupations; set down natural history and all its progeny as a deformed and abortive creation; and entertained the greatest disdain for a would-be science, which could never even step within the threshold of real knowledge. In this mood of mind I betook myself to the mathematics, and the branches of study appertaining to that science, as being built upon secure foundations, and so worthy of my consideration.

- Mary Shelley, Frankenstein

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

1. Vector
2. Scalar
3. Addition (of vectors)
4. Scalar multiplication.
5. Length (or magnitude) of a vector.
6. Standard basis vectors (defined by a particular coordinate system).
7. Dot product.
8. Orthogonal.
9. \(\text{proj}_{\vec{a}}\vec{b}\).
10. \(\text{orth}_{\vec{a}}\vec{b}\).
11. Cross product.
12. Scalar triple product.

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

1. Properties of vector addition and scalar multiplication.
2. Formula for the magnitude of a vector given in component form.
3. Properties of the dot product.
4. Formula for the dot product in component form.
5. Formulas for \(\text{proj}_{\vec{a}}\vec{b}\) and \(\text{orth}_{\vec{a}}\vec{b}\).
6. Properties of the cross product.
7. Formula for the cross product in component form.
8. Relationship between cross products and areas.
9. Associativity of the scalar triple product (equation (6) on page 660).
10. Formula for the scalar triple product (equation (7)).
11. Relationship between scalar triple products and volume.

Solve the following problems:

1. Section 9.2, problems 6, 7, 11. 15, and 19.
2. Section 9.3, problems 3, 7, 11, 17, 21, and 31.
3. Section 9.4, problems 1, 3, 7, 15, 20, and 27.

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