Refer to vectors $\vec{a}$ and $\vec{b}$ in the figure mentioned.

On another copy of the figure, draw $-\vec{b}$, the opposite of vector $\vec{b}$, as a position vector (starting at the origin). Then translate $-\vec{b}$ so that its tail is at the head of $\vec{a}$. Using the definition of vector addition, draw vector $\vec{a}+(-\vec{b})$. Explain why $\vec{a}-\vec{b}$ is equivalent to $\vec{a}+(-\vec{b})$.

For the following problem, draw the position vector on graph paper. Show the circumscribed "box" that makes the vector look three-dimensional. Write the coordinates of the point at the head of the vector.

$$\vec{p}=3 \vec{i}+8 \vec{j}+4 \vec{k}$$

For the following problem, find a particular equation of the plane described.

Perpendicular to $\vec{n}=3 \vec{i}-5 \vec{j}+4 \vec{k}$, containing the point (6,-7,-2)

For the following problem, use the definition of dot product to find $\vec{a} \cdot \vec{b}$, where $\theta$ is the angle between $\vec{a}$ and $\vec{b}$ when they are placed tail-to-tail.

$|\vec{a}|=29,|\vec{b}|=50$, and $\theta=127^{\circ}$

What is the major difference in meaning between $\vec{a} \cdot \vec{b}$ and $\vec{a} \times \vec{b}$?

Find the point on the line in the previous problem for which d = 34.

For the following problem, find the direction cosines and direction angles for the position vector to the given point.

(2,-5,3)

For the following problem, find the cross product using determinants.

$$(4 \vec{i}-3 \vec{j}-\vec{k}) \times(2 \vec{i}-\vec{j}+\vec{k})$$

Write the coordinates of the point 40% of the way from the head of $\vec{a}$ to the head of $\vec{b}$.

The speeds in the previous part are in feet per second. Thus, the position vectors of the planes are

Flight 007:

$$\vec{r}_1= (3000-100 t) \vec{i}+(2000+50 t) \vec{f} +(0+20 t) \vec{k}$$

Flight 1776:

$$\vec{r}_2= (1000+40 t) \vec{i}+(500+200 t) \vec{j} +(300-15 t) \vec{k}$$

where the parameter t is time in seconds. Write the displacement vector from Flight 1776 to Flight 007 as a function of time. Using appropriate algebraic or numerical techniques, find the time t at which the flights were closest to each other by finding the value of t at which the length of the displacement vector was a minimum. Explain why the closest the flights came to each other is not the same as the closest the two paths came to each other. Is there cause for concern about how close the flights came to each other?

For the following problem, given the equation of a line, complete the given part

$$\vec{r}=\left(6+\frac{1}{9} d\right) \vec{i}+\left(7+\frac{8}{9} d\right) \vec{i}+\left(-5+\frac{4}{9} d\right) \vec{k}$$

Confirm that $\vec{u}$ really is a unit vector.

Find the position vector for the point 40% of the way from the head of $\vec{a}$ to the head of $\vec{b}$.

Find a displacement vector from the head of $\vec{a}$ to a point 40% of the way from the head of $\vec{a}$ to the head of $\vec{b}$.