Prove that if $\vec{n}=A \vec{i}+B \vec{j}+C\vec{k}$ is a normal vector to a plane, then a particular equation of the plane is Ax + By + Cz = D, where D stands for a constant.

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, 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.

For the following problem, find the area.

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

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 $\vec{b}=3 \vec{i}-4 \vec{j}+12 \vec{k}$

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

2/3 of the way from (7,8,11) to (34,32,14)

Prove that these two planes are perpendicular.

$$\begin{aligned} & 2 x-5 y+3 z=10 \\ & 7 x+4 y+2 z=17 \end{aligned}$$