Exam 12: Vectors and Matrices

An airplane is flying at an airspeed of 620 km/hr in a crosswind blowing from the southeast at a speed of 55 km/hr. To end up going due east, the plane should head ---------°south of east and will have a speed of ---------- km/hr relative to the ground. Round each answer to 2 decimal places.

Let the vector $\vec{t}=\left(x_{1}, y_{1}, x_{2}, y_{2}, x_{3}, y_{3}\right)$ describe a triangle with vertices at $\left(x_{1}, y_{1}\right)$ , $\left(x_{2}, y_{2}\right)$ , and $\left(x_{3}, y_{3}\right)$ . If the triangle $\vec{u}=(4,2,6,2,4,4)$ is rotated through an angle of $30^{\circ}$ clockwise about the origin, what is the resulting vector? Round each entry to 2 decimal places.

Which of the following vectors are parallel to $\vec{v}=5 \vec{i}-2 \vec{j}+2 \vec{k}$ ? Mark all that apply.

If $\vec{v}=-2 \vec{i}+2 \vec{j}-3 \vec{k}$ . Find $c$ such that $\vec{w}=-6 \vec{i}+6 \vec{j}+c \vec{k}$ is parallel to $\vec{v}$ .

The figure below shows the vector $\vec{v}=$ ---------- $\vec{i}+$ ---------- $\vec{j}$ .

Three people stand in the middle of a field. The first person walks 15 yards north and then 15 yards east. The second person remains where he is. The third person walks 20 yards north and then 9 yards east. The angle formed by drawing a line from the first person to the second person to the third person is ----------°. Round to the nearest whole number.

For $\vec{q}=3 \vec{i}+\vec{j}-4 \vec{k}$ and $\vec{r}=-2 \vec{i}-2 \vec{j}+\vec{k}$ , $(\vec{r} \cdot \vec{r}) \vec{q}=\ldots \vec{i}+\ldots \vec{j}+\ldots \vec{k}$ .

Let the $x$ -axis point east, the $y$ -axis point north, and the $z$ -axis point upward, and let the unit vectors be in kilometers. What does the vector $2 \vec{i}-6 \vec{j}+0.5 \vec{k}$ represent?

For what value of $x$ are $5 \vec{i}-6 \vec{j}+2 \vec{k}$ and $x \vec{i}-2 \vec{j}+2 \vec{k}$ perpendicular?

A boat in the ocean at point $(39,58,0)$ is looking at the light on a lighthouse at point $(39,0,25)$ . The light house is 3 units high, and sits at the edge of a cliff above the ocean. The displacement vector between the boat and the base of the lighthouse is ------------- $\vec{i}+$ ---------- $\vec{j}+$ ---------- $\vec{k}$ .

Let $\mathbf{A}=\left(\begin{array}{ccc}1 & 2 & 0 \\ -1 & 4 & -2 \\ 0 & 2 & 3\end{array}\right)$ and $\mathbf{B}=\left(\begin{array}{ccc}4 & 3 & -2 \\ 0 & 4 & 0 \\ -1 & 1 & -1\end{array}\right)$ a) Find $\mathbf{A}+\mathbf{B}$ b) If $\vec{u}=(1,2,3)$ , find $\mathbf{A} \vec{u}$ . c) If $\vec{v}=(-1,0,1)$ , find $\overrightarrow{\mathbf{B}} \vec{v}$ .

A canoe can travel $16 \mathrm{ft} \mathrm{sec}$ in still water. Suppose a river is flowing at a rate of 5 $\mathrm{ft} \mathrm{sec}$ . If the canoe is traveling downstream at an angle of $45^{\circ}$ with the current, what is the speed of the canoe? Round numbers to 3 decimal places if necessary.

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 $\vec{P}_{\text {new }}=\mathbf{T} \vec{P}_{\text {old }}=\left(\begin{array}{ccc}0.95 & 0.11 & 0.10 \\ 0.04 & 0.85 & 0.15 \\ 0.01 & 0.04 & 0.75\end{array}\right) \vec{P}_{\text {old }}$ . What percent of the members of the second party switch to the first party each year?

Do the points $(11,19),(13,15),(17,17)$ , and $(15,21)$ form a square?

Given the vector $\vec{N}=\{1,5,9\}$ find vector $\vec{B}=2 \vec{N}$ .

If $\vec{A}=(1,2,-1,3)$ and $\vec{B}=(4,8,2,0)$ , find $\vec{C}=3 \vec{A}-5 \vec{B}$

Which of the following would be described by vectors rather than scalars?

If $\vec{\alpha}=(1.1,3.7,4.1,5.6)$ and $\beta=(10.5,10.7,10.8,11.1)$ , find $\vec{\gamma}=6.5(\vec{\alpha}-\vec{\beta})$ .

A vector of length 7 that points in the same direction as $2 \vec{i}+\vec{j}-6 \vec{k}$ is $\vec{i}+\ldots \vec{j}+\ldots \vec{k}$ . Give each answer to 3 decimal places.

A kite on a 26 meter string is flying with an angle of $33^{\circ}$ with the ground. What is the magnitude and the direction of the vector from the kite to the ground. Round numbers to 3 decimal places if necessary.