Exercise 21

Recall that given any vector $\vv$, we can calculate its length, $\vecmag{\vv}$. Also, we say that two vectors that are (non-zero) scalar multiples of one another are parallel.

1. Let $\vv = \langle 3,4 \rangle$ in $\R^2$. Compute $\vecmag{\vv}$, and determine the components of the vector $\vu = \frac{1}{\vecmag{\vv}} \vv$. What is the magnitude of the vector $\vu$? How does its direction compare to $\vv$?

2. Let $\vw = 3\vi - 3\vj$ in $\R^2$. Determine a unit vector $\vu$ in the same direction as $\vw$.

3. Let $\vv = \langle 2, 3, 5 \rangle$ in $\R^3$. Compute $\vecmag{\vv}$, and determine the components of the vector $\vu = \frac{1}{\vecmag{\vv}} \vv$. What is the magnitude of the vector $\vu$? How does its direction compare to $\vv$?

4. Let $\vv$ be an arbitrary nonzero vector in $\R^3$. Write a general formula for a unit vector that is parallel to $\vv$.