Exam 12: Vectors and Matrices

In the figure below, each square is 6 units along each side. In component form, $\vec{u}=$ ------------- $\vec{i}+$ ---------- $\vec{j}$ .

Let $\mathbf{A}=\left(\begin{array}{cc}4 & 5 \\ -1 & 0\end{array}\right), \mathbf{B}=\left(\begin{array}{ccc}1 & -1 & 2 \\ 3 & 4 & 0 \\ -2 & 1 & 8\end{array}\right), \vec{u}=(1,2)$ , and $\vec{v}=(1,2,3)$ . Which of the Following are defined.

For $\vec{q}=-3 \vec{i}+\vec{j}-3 \vec{k}$ and $\vec{r}=-2 \vec{i}-3 \vec{j}+\vec{k}$ , what is $((\vec{r} \cdot \vec{q}) \vec{r}) \cdot \vec{q}$ ?

How many ft-lbs of work are required to lift a 371 pound refrigerator 3 inches straight up? Round to 2 decimal places.

A car travels 19 miles south and then 23 miles east. Which of the following is true?

If $\|\vec{v}\|=1.3$ , what is $\|-\vec{v}\|$ ?

Let the student vector $\vec{P}=(120,126,117,121)$ 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 $\vec{R}$ ?

Let the student vector $\vec{P}=(123,129,119,118)$ be the number of ninth, tenth, eleventh, and twelfth graders in a high school. If the number of students in each grade doubles, what is the new student vector $\vec{R}$ ?

Given the vectors $\vec{N}=\{2,5,8\}$ and $\vec{M}=\{7,9,11\}$ find vector $\vec{B}=\vec{N} \bullet \vec{M}$ .

In the figure below, each square is 9 units along each side. In component form,

Does $-3 \vec{v}+5 \vec{v}$ point in the same direction as $\vec{v}$ , the opposite direction, or neither?

Jack and Jill begin walking away from a water well. Jill walks 2 meters north, and then 12 meters east. When Jill stops, Jack is twice as far from the well as she is. How far is Jack from the well? Round numbers to 3 decimal places if necessary.

Jack and Jill begin walking away from a water well. Jill walks 4 meters west, and then 2 meters north. When Jill stops, Jack is three times as far from the well as she is, but in the opposite direction. If $\vec{v}$ is the vector pointing from Jack to the well, find the length and direction of $\vec{v}$ . Round numbers to 3 decimal places if necessary.

Given $\mathbf{R}=\left(\begin{array}{cc}1 & -4 \\ 1 & 5\end{array}\right)$ and $\vec{u}=(1,-4)$ , does $\mathbf{R} \vec{u}=(17,-19)$ ?

Find the length of the vector $-2 \vec{v}+4 \vec{w}$ if $\vec{v}=\vec{i}+2 \vec{j}-3 \vec{k}$ and $\vec{w}=-2 \vec{j}$ .

For what value of $x$ do $6 \vec{i}-x \vec{j}+(x-2) \vec{k}$ and $x \vec{i}-(x+1) \vec{j}+3 \vec{k}$ have the same length?

Let $\mathbf{A}=\left(\begin{array}{cc}1 & 2 \\ 0 & -1\end{array}\right)$ and $\vec{u}=(5,-1)$ . Find $\vec{w}$ such that $\vec{u}=\mathbf{A} \cdot \vec{w}$ .

The rectangle with vertices $(1,0),(1,4),(2,0)$ , and $(2,4)$ is rotated through an angle of $21^{\circ}$ about the origin. What are the coordinates of the new rectangle? Round numbers to 3 decimal places if necessary.

A cannon is fired at a $15^{\circ}$ angle with the ground at a speed of $81 \mathrm{ft} / \mathrm{sec}$ . Resolve the velocity vector into horizontal and vertical components.

Let $\mathbf{A}=\left(\begin{array}{cc}1 & 2 \\ 0 & -1\end{array}\right)$ and $\vec{u}=(4,1)$ . Find $\vec{w}$ such that $\vec{w}=\mathbf{A} \cdot \vec{u}$ .