A bullet has pierced the wall and ceiling of a small house, and it may have lodged in the roof. You have studied vectors, so the investigators call on you to calculate the point in the roof where the bullet is expected to be found. The dimensions of the house are in feet.

You set up a three-dimensional coordinate system as shown, with the origin at the floor in the back corner of the house. The bullet pierced the wall at the point (10,14,3) and then pierced the ceiling at the point (7,18,8). Find a unit vector in the direction of the bullet's path.

For the following problem, find $\vec{a} \cdot \vec{b}$ and the angle between $\vec{a}$ and $\vec{b}$ when they are tail-to-tail.

$$\begin{aligned} \vec{a}&=-3 \vec{i}+5 \vec{j}+2 \vec{k}\\ \vec{b}&=6 \vec{i}-3 \vec{j}+\vec{k} \end{aligned}$$

For the following problem, find the position vector of the indicated point.

130% of the way from (2,9,7) to (4,-3,1)

For the following problem, find the area.

The triangle with vertices (3,7,5), (2,-1,7), and (-4,6,10)

For the following problem, prove that the vector is a unit vector, and find its direction cosines and direction angles.

$$\frac{1}{9} \vec{i}-\frac{4}{9} \vec{J}+\frac{8}{9} \vec{k}$$

For the following problem, find the point where the given line intersects the given plane.

Line: $\vec{r}=\left(4+\frac{1}{3} d\right) \vec{t}+\left(1+\frac{2}{3} d\right)\vec {j}+\left(7+\frac{2}{3} d\right) \vec{k}.$

Plane: $x+4 y-3 z=35$

For the following problem, find $\vec{a} \cdot \vec{b}$ and the angle between $\vec{a}$ and $\vec{b}$ when they are tail-to-tail.

$$\begin{aligned} \vec{a}&=3 \vec{i}+2 \vec{j}-4 \vec{k}\\ \vec{b}&=8 \vec{i}+5 \vec{j}-2 \vec{k} \end{aligned}$$

For the following problem, find the area.

The parallelogram determined by $\vec{c}=4 \vec{i}+4 \vec{\jmath}-7 \vec{k}$ and $\vec{d}=-2 \vec{i}+5 \vec{j}-14 \vec{k}$

For the following problem, use the techniques of the previous problems to find the vector or point.

Find the midpoint of the segment connecting points E(6,2) and F(10,-4). From the result, give a quick way to find the midpoint of the segment connecting two given points.

The following problem refer to the vectors

$$\begin{aligned} & \vec{a}=5 \vec{i}+2 \vec{j}+9 \vec{k} \\ & \vec{b}=3 \vec{i}+8 \vec{i}+4 \vec{k} \end{aligned}$$

Find the angle between $\vec{a}$ and $\vec{b}$ if they are placed tail-to-tail.

Find the value of A that makes these two planes perpendicular.

$$\begin{aligned} & A x+3 y-2 z=-8 \\ & 4 x-5 y+z=7 \end{aligned}$$

For the following problem, find the position vector of the indicated point.

1/3 of the way from (5,1,23) to (26,13,14)

For the following problem, find $\vec{a} \cdot \vec{b}$ and the angle between $\vec{a}$ and $\vec{b}$ when they are tail-to-tail.

$$\begin{aligned} \vec{a}&=2 \vec{i}+5 \vec{j}+3 \vec{k}\\ \vec{b}&=7 \vec{i}-\vec{j}+4 \vec{k} \end{aligned}$$

For the following problem, find the area.

The parallelogram determined by $\vec{a}=2 \vec{i}+3 \vec{j}+6 \vec{k}$ and $\vec{b}=3 \vec{i}-4 \vec{j}+12 \vec{k}$