Vectors

After being bored in your previous calculus class (and definitely not in multivariable calculus) you count the floor tiles to see how large the room is. You now know that the room is 32 feet wide and 42 feet in length. Additionally, your tall friend Chuck can barely jump and touch the ceiling which means the ceiling is 10 feet high. Your calculus professor notices you doing these measurements and…

1. Let $\vu = \langle 2,3\rangle$ and $\vv = \langle -1,2\rangle$. Find $\vecmag{\vu}$, $\vecmag{\vv}$, and $\vecmag{\vu+\vv}$. Is it true that $\vecmag{\vu+\vv} = \vecmag{\vu}+\vecmag{\vv}$?

2. Under what conditions will $\vecmag{\vw_1+\vw_2} = \vecmag{\vw_1}+\vecmag{\vw_2}$?

3. With the vector $\vu = \langle 2,3\rangle$, find the lengths of $2\vu$, $3\vu$, and $-2\vu$, respectively, and use proper notation to label your results.

4. In general, if $t$ is any scalar, how will $\vecmag{t \vw}$ be related to $\vecmag{\vw}$?

5. Of the vectors $\vi$, $\vj$, and $\vi+\vj$, which are unit vectors?

6. Find a unit vector $\vv$ whose direction is the same as $\vu = \langle -2, 3\rangle$.

7. Find a unit vector $\vv$ in the opposite direction…

Let $\vv = \langle 1, -2 \rangle$, $\vu = \langle 0, 4 \rangle$, and $\vw = \langle -5, 7 \rangle$.

1. Determine the components of the vector $\vu - \vv$.

2. Determine the components of the vector $2\vv - 3\vu$.

3. Determine the components of the vector $\vv + 2\vu - 7 \vw$.

4. Determine scalars $a$ and $b$ such that $a \vv + b\vu = \vw$. Show all of your work in finding $a$ and $b$.

Let $\vu = \langle 2, 1 \rangle$ and $\vv = \langle 1, 2 \rangle$.

1. Determine the components and draw geometric representations of the vectors $2\vu$, $\frac{1}{2}\vu$, $(-1)\vu$, and $(-3)\vu$ on the same set of axes.

2. Determine the components and draw geometric representations of the vectors $\vu + \vv$, $\vu + 2\vv$, and $\vu + 3\vv$ on the same set of axes.

3. Determine the components and draw geometric representations of the vectors $\vu - \vv$, $\vu - 2\vv$, and $\vu - 3\vv$ on the same set of axes.

4. Recall that $\vu - \vv = \vu + (-1)\vv$. Sketch the vectors $\vu$, $\vv$, $\vu+\vv$, and $\vu-\vv$ on…

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 deter…