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:[edit]
- Vector
Definition: A geometric quantity with magnitude (length) and direction. Represented numerically as an ordered list of n-tuples, where n is the dimension of space in which the vector "resides" (e.g., in 3 dimensions, or \(R^3\), a vector would be represented as \(<x_1, y_1, z_1>\))
Example: \(\vec{v}=<1,2,3>\)
Non-Example: \(\vec{v}=17\)
- Scalar
Definition: A numerical quantity.
Example: \(x = 3\)
Non-Example: \(x=monkey\)
- Addition (of vectors)
Definition:
Example:
Non-Example:
- Scalar multiplication.
Definition:
Example:
Non-Example:
- Length (or magnitude) of a vector.
Definition:
Example:
Non-Example:
- Standard basis vectors (defined by a particular coordinate system).
Definition:
Example:
Non-Example:
- Dot product.
Definition:
Example:
Non-Example:
- Orthogonal.
Definition:
Example:
Non-Example:
- \(\text{proj}_{\vec{a}}\vec{b}\).
Definition:
Example:
Non-Example:
- \(\text{orth}_{\vec{a}}\vec{b}\).
Definition:
Example:
Non-Example:
- Cross product.
Definition:
Example:
Non-Example:
- Scalar triple product.
Definition:
Example:
Non-Example:
Carefully state the following theorems (you do not need to prove them):[edit]
- Properties of vector addition and scalar multiplication.
- Formula for the magnitude of a vector given in component form.
- Properties of the dot product.
- Formula for the dot product in component form.
- Formulas for \(\text{proj}_{\vec{a}}\vec{b}\) and \(\text{orth}_{\vec{a}}\vec{b}\).
- Properties of the cross product.
- Formula for the cross product in component form.
- Relationship between cross products and areas.
- Associativity of the scalar triple product (equation (6) on page 660).
- Formula for the scalar triple product (equation (7)).
- Relationship between scalar triple products and volume.
Solve the following problems:[edit]
- Section 9.2, problems 6, 7, 11. 15, and 19.
- Section 9.3, problems 3, 7, 11, 17, 21, and 31.
- Section 9.4, problems 1, 3, 7, 15, 20, and 27.
