Exam 12: Vectors and Matrices

Let $\vec{A}=(3,2,7,2)$ and $\vec{B}=(6,8,3,5)$ . Find $\vec{A}-3 \vec{B}$ .

The unit vector from the point $(4,3)$ toward the point $(5,5)$ has its head at the point ( -------------,------------). Round to 2 decimal places.

A man leaves his car and walks 2 miles northeast, 4 miles east, and then 8 miles southwest. How far is the person from his car? In what direction must he walk to head directly to his car? Round numbers to 3 decimal places if necessary.

Perform the computation $3(3 \vec{i}-\vec{j})+5 \vec{j}$ .

The inverse of a matrix $\mathbf{A}$ , denoted by $\mathbf{A}^{-1}$ , is such that if $\vec{v}=\mathbf{A} \vec{u}$ , then $\vec{u}=\mathbf{A}^{-1} \vec{v}$ . For a matrix $\mathbf{A}=\left(\begin{array}{ll}a & b \\ c & d\end{array}\right)$ , the inverse $\mathbf{A}^{-1}$ is given by $\mathbf{A}^{-1}=\frac{1}{D}\left(\begin{array}{cc}d & -b \\ -c & a\end{array}\right)$ , where $D=a d-b c \mathbf{A}^{-1}$ is undefined if $D=0 .$ Let $\mathbf{A}=\left(\begin{array}{cc}1 & 1 \\ 2 & -2\end{array}\right)$ . Does $\mathbf{A}^{-1}=\frac{1}{-4}\left(\begin{array}{cc}-2 & -2 \\ -1 & 1\end{array}\right) ?$

A snow cone stand sells three sizes of snow cones: small, medium, and large. Let $\vec{N}=(s, m, l)$ give the number of each type of cone sold in one day. Let $\vec{P}=\left(p_{s}, p_{m}, p_{l}\right)$ give the price charged for each size of snow cone, $\vec{C}=\left(c_{s}, c_{m}, c_{l}\right)$ give the cost of making each size of snow cone, and $M=\left(m_{s}, m_{m}, m_{l}\right)$ give the maximum number of each size that can be sold (because of the number of each size cup on hand.) Let $\vec{T}=\vec{P}-\vec{C}$ . Which of the following gives the total profit earned in one day?

For what value of $x$ are $16 \vec{i}-12 \vec{j}+4 \vec{k}$ and $x \vec{i}-(x-1) \vec{j}+\vec{k}$ parallel?

Let $\vec{P}=(o, r)$ be the population vector of a city giving $o$ , the number of people who own their home and $r$ , the number of people who rent. Suppose that each year, 3\% of the people who own their home move to a rental and $15 \%$ of the people who rent purchase a home and become owners. Let $\mathbf{A}=\left(\begin{array}{ll}a_{11} & a_{12} \\ a_{21} & a_{22}\end{array}\right)$ be such that $\vec{P}_{\text {new }}=\mathbf{A} \vec{P}_{\text {old }}$ . What is $a_{11}$ ?

If $R=(2,3,4,5,6)$ and $S=(0,1,4,5,9)$ , then what is $\vec{A}=4 \vec{R}-3 \vec{S}$ ?

Let $P=(6,2,1)$ and $Q=(-2,0,5)$ . Write the displacement vector $\overrightarrow{P Q}$ in component form.

In the figure below, each square is 7 units along each side. The vector perpendicular to the displacement vector $\overrightarrow{R S}$ is ----------- $\vec{i}+\ldots \vec{j}$ .

For $\vec{q}=3 \vec{i}+\vec{j}-5 \vec{k}$ and $\vec{r}=3 \vec{i}-5 \vec{j}+\vec{k}$ , what is $\vec{q} \cdot \vec{r}$ ?

How many ft-lbs of work are done pushing a $15-\mathrm{lb}$ stroller with a $25 \mathrm{lb}$ toddler in it up a 45 yard hill if the slope of the hill is $15^{\circ}$ ? Round to 1 decimal place.

A country has two main political parties. The vector $\vec{P}=(f, s, n)$ gives the number of people who are members of the first party, the second party, and neither party, respectively. Suppose $\vec{P}_{\text {new }}=\mathbf{T} \vec{P}_{\text {old }}=\left(\begin{array}{lll}0.80 & 0.11 & 0.10 \\ 0.17 & 0.85 & 0.15 \\ 0.03 & 0.04 & 0.75\end{array}\right) \vec{P}_{\text {old }}$ . Let $\vec{P}_{2005}=(25,30,18)$ be the population vector (in tens of thousands) in the year 2005. What is $\vec{P}_{2006}$ ?

A cyclist goes at 6 mph due north and feels the wind coming against him at a relative velocity of 3 mph due west. The actual velocity of the wind is ----------mph at an angle of ---------------°west of north. Round both answers to 1 decimal place.

Let $\vec{A}=(7,4,7,2)$ and $\vec{B}=(4,9,3,5)$ . Find $\frac{\vec{A}}{2}+\vec{B}$

An airplane flies at an airspeed of $500 \mathrm{~km} / \mathrm{hr}$ in a cross-wind that is blowing from the southwest at a speed of $29 \mathrm{~km} \ \mathrm{hr}$ . What direction should the plane fly to end up going due south?

As $t$ changes, what happens to the tip of the vector $\vec{r}=(1+4 t) \vec{i}+(2-3 t) \vec{j}$ ?

Let $\vec{v}$ be a vector of length 3 pointing $30^{\circ}$ north of east. Find the length and direction of $-4 \vec{v}$ .

The angle between the vectors $5 \vec{i}-5 \vec{j}+\vec{k}$ and $4 \vec{i}+4 \vec{j}-\vec{k}$ is---------. Round to the nearest whole number.