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

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.

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

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 a unit vector in the direction of $\vec{b}$.

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

$$\frac{7}{9} \vec{i}+\frac{4}{9} \vec{j}-\frac{4}{9} \vec{k}$$

Given an L-shaped house that is to be built. Roof 1 and Roof 2 will have normal vectors

$$\begin{aligned} & \vec{n} 1=0 \vec{i}+6 \vec{j}+12 \vec{k} \\ & \vec{n} 2=6 \vec{i}+0 \vec{j}+12 \vec{k} \end{aligned}$$

The two roofs will meet at a "valley." Point (30,30,10) is at the lower end of the valley. The dimensions are in feet.

Why do you think builders put flashing in roof valleys?

Given an L-shaped house that is to be built. Roof 1 and Roof 2 will have normal vectors

$$\begin{aligned} & \vec{n} 1=0 \vec{i}+6 \vec{j}+12 \vec{k} \\ & \vec{n} 2=6 \vec{i}+0 \vec{j}+12 \vec{k} \end{aligned}$$

The two roofs will meet at a "valley." Point (30,30,10) is at the lower end of the valley. The dimensions are in feet.

The two roof sections form a dihedral angle equal to the angle between the two normal vectors or to the supplement of this angle. The flashing must be bent to fit this angle. Calculate the obtuse dihedral angle between the two roof sections.

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

Find the position vector of the point $\frac{2}{3}$ of the way from point C(11,5) to point D(2,17).

Given an L-shaped house that is to be built. Roof 1 and Roof 2 will have normal vectors

$$\begin{aligned} & \vec{n} 1=0 \vec{i}+6 \vec{j}+12 \vec{k} \\ & \vec{n} 2=6 \vec{i}+0 \vec{j}+12 \vec{k} \end{aligned}$$

The two roofs will meet at a "valley." Point (30,30,10) is at the lower end of the valley. The dimensions are in feet.

A piece of sheet metal flashing is to be fitted into the valley to go underneath the shingles. How long will the valley be?