Exam 12: Vectors and Matrices

Given $\mathbf{S}=\left(\begin{array}{cc}1 & -4 \\ -2 & 7\end{array}\right), \vec{u}=(1,0)$ , and $\vec{v}=(-4,2)$ , what is $\mathbf{S}(\vec{u}+\vec{v})$ ?

Let $\mathbf{A}=\left(\begin{array}{cc}2 & 1 \\ -3 & -1\end{array}\right)$ . Find $\mathbf{A}^{-1}$ .

Let the student vector $\vec{P}=(121,129,119,119)$ be the number of ninth, tenth, eleventh, and twelfth graders in a high school. If the number of students in each grade increases by $5 \%$ , what is the new student vector $\vec{R}$ ? Round to the nearest whole number.

Find the length of the vector $\vec{v}=-2.5 \vec{i}+3.5 \vec{j}-4.1 \vec{k}$ to 3 decimal places.

A gun mounted on a train points vertically upward. The train moves horizontally due east at 80 mph and the gun fires a bullet with a muzzle velocity of 80 mph. The speed and direction of the bullet relative to the ground is ------------ mph toward the east at ------------° to the horizontal. Round the first answer to 2 decimal places.

What properties of vector addition and scalar multiplication are necessary to show $(5+a)(\vec{v}+\vec{w})=5 \vec{w}+a \vec{w}+5 \vec{v}+a \vec{v}$

A snow cone stand sells three sizes of snow cones: small, medium, and large. Let $\vec{N}=(80,95,97)$ give the number of each type of cone sold in one day. Let $\vec{P}=(1.50,2.00,2.50)$ give the price (in dollars) charged for each size of snow cone, $\vec{C}=(0.30,0.40,0.50)$ give the cost (also in dollars) of making each size of snow cone, and $M=(150,200,200)$ give the maximum number of each size that can be sold (because of the number of each size cup on hand). How much money was spent making the medium snow cones that day?

The vectors $\vec{u}=4 \vec{i}+5 \vec{j}-6 \vec{k}$ and $\vec{v}=-12 \vec{i}-15 \vec{j}+18 \vec{k}$ are parallel.

Simplify $12 \vec{v}-3(4 \vec{v})$

Let $\vec{P}=\left(p_{1}, p_{2}, p_{3}\right)$ give the profit a salesman makes from selling three different models of computers. Let $\vec{S}=\left(s_{1}, s_{2}, s_{3}\right)$ give the number of each computer model sold each week. What does $\vec{P} \cdot \vec{S}$ represent?

Simplify the following: $3(2 \vec{v}+7 \vec{w})-(-3 \vec{v}-7 \vec{w})$

Given $\mathbf{R}=\left(\begin{array}{cc}-3 & -4 \\ 1 & 5\end{array}\right), \mathbf{S}=\left(\begin{array}{cc}1 & 3 \\ -2 & 7\end{array}\right)$ , and $\vec{v}=(-1,2)$ what is $(\mathbf{R}-\mathbf{S}) \vec{v}$ ?

In certain cases, there is a nonzero vector $\vec{v}$ and a scalar $\lambda$ for a matrix $\mathbf{A}$ such that $\mathbf{A} \vec{v}=\lambda \vec{v}$ . The vector $\vec{v}$ is called an eigenvector of $\mathbf{A}$ with eigenvalue $\lambda$ . Let $\mathbf{A}=\left(\begin{array}{cc}1 & 2 \\ 6 & -3\end{array}\right)$ with eigenvector $\vec{v}=(3,-9)$ . What is its eigenvalue?

A man is sitting 9 meters above the ground in a tree which is 10 meters directly south of the fence corner of a field. A bird is spotted 2 meters above the ground and 8 meters directly north of the fence corner. How far is the man from the bird? Round your answer to 3 decimal places.

There are six people taking a vocational exam with both oral and written parts. Their scores (out of 100) on the written section are given by the vector $\vec{r}=(71,49,87,98,81,63)$ . Their scores (out of 100) on the oral section are given by the vector $\vec{s}=(79,74,82,95,97,66)$ . Find the vector giving their composite score if the written part counts twice as much as the oral part.

Given that $\mathbf{A}=\left(\begin{array}{ccc}1 & 1 & 2 \\ 4 & -1 & -3 \\ 4 & 0 & 2\end{array}\right)$ and $\mathbf{B}=\left(\begin{array}{ccc}1 & 1 & -3 \\ 2 & -4 & 0 \\ -1 & -2 & 3\end{array}\right)$ , let $((\mathbf{A}-\mathbf{B})-\mathbf{B})-\mathbf{B}=\mathbf{C}$ , where $\mathbf{C}=\left(\begin{array}{lll}c_{11} & c_{12} & c_{13} \\ c_{21} & c_{22} & c_{23} \\ c_{31} & c_{32} & c_{33}\end{array}\right)$ . What is $c_{12}$ ?

A man is sitting 8 meters above the ground in a tree which is 10 meters directly south of the fence corner of a field. A bird is spotted 3 meters above the ground and 9 meters directly north of the fence corner. In what direction must the man face to look directly at the bird? Round your answer to 3 decimal places.

Suppose a man starts at work and drives $8 \mathrm{~km}$ due south, and then drives $13 \mathrm{~km}$ southeast. How far is the man from work? Round numbers to 3 decimal places if necessary.

A retailer's total monthly sales of three different models of television is given by the vector $\vec{S}=(12,25,12)$ . If the sales for each model go down by $15 \%$ the next month, what is $\vec{Q}$ , the next month's total sales? Round entries to the nearest whole number.