Vectors; Lines, Planes, and Quadric Surfaces in Space

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10.2 Assess Your Understanding
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Concepts and Vocabulary

True or False A vector is an object that has magnitude and direction.

True

Scalars are quantities that have only ______.

Magnitude

The product of a scalar $a$ and a vector $\mathbf{v}$ is called a(n) _____ ______ of $\mathbf{v}$ .

Scalar multiple

Multiple Choice The vectors $-\mathbf{v}$ and $\mathbf{v}$ have [(a) the same, (b) different] magnitude and the [(c) same, (d) opposite] direction.

(a); (d)

Skill Building

State which of the following are scalars and which are vectors:
1. (a) Volume
2. (b) Speed
3. (c) Force
4. (d) Work
5. (e) Mass
6. (f) Distance
7. (g) Age
8. (h) Velocity

Scalars: (a), (b), (d), (e), (f), (g). Vectors: (c), (h)

In Problems 6–14, use the vectors in the above figure to graph each of the following vectors:

$2\mathbf{v}$

$-2\mathbf{v}$

$\mathbf{v} + \mathbf{w}$

$\mathbf{v}-\mathbf{w}$

$\mathbf{w}-\mathbf{v}$

$\mathbf{v}-2\mathbf{w}$

$(\mathbf{v+w}) +3\mathbf{u}$

$\mathbf{v}+( \mathbf{w}+3\mathbf{u})$

$2\,\mathbf{u}-\dfrac{1}{3}(\mathbf{v}-\mathbf{w})$

In Problems 15–22, use the vectors in the figure below.

Find the vector $\mathbf{x}$ if $\mathbf{x}+\mathbf{B}=\mathbf{F}$ .

$\mathbf{x}=\mathbf{A}$

Find the vector $\mathbf{x}$ if $\mathbf{x}+\mathbf{K}= \mathbf{C}$ .

703

Write $\mathbf{C}$ in terms of $\mathbf{E}$ , $\mathbf{D}$ , and $\mathbf{F}$ .

${\bf C}=-{\bf D} - {\bf E} - {\bf F}$

Write $\mathbf{G}$ in terms of $\mathbf{C}$ , $\mathbf{D}$ , $\mathbf{E}$ , and $\mathbf{K}$ .

Write $\mathbf{E}$ in terms of $\mathbf{G}$ , $\mathbf{H}$ , and $\mathbf{D}$ .

$\mathbf{E}=\mathbf{G}+\mathbf{H}-\mathbf{D}$

Write $\mathbf{E}$ in terms of $\mathbf{A}$ , $\mathbf{B}$ , $\mathbf{C}$ , and $\mathbf{D}$ .

What is $\mathbf{A}+\mathbf{B}+\mathbf{K}+\mathbf{G}$ ?

$\mathbf{0}$

What is $\mathbf{A}+\mathbf{B}+\mathbf{C}+\mathbf{H}+\mathbf{G}$ ?

Applications and Extensions

Resultant Force Two forces $\mathbf{F}_{1}$ and $\mathbf{F}_{2}$ act on an object shown in the figure. Graph the vector representing the resultant force; that is, find $\mathbf{F}_{1}+\mathbf{F}_{2}.$

Resultant Force Two forces $\mathbf{F}_{1}$ and $\mathbf{F} _{2}$ act on an object shown in the figure. Graph the vector representing the resultant force; that is, find $\mathbf{F}_{1}+\mathbf{F}_{2}.$

Air Travel An airplane is flying due north at a constant airspeed of 560 $\rm{mi/h}$ . There is a wind $75 \,\rm{mi/h}$ blowing from the east.
1. (a) Draw vectors representing the velocities of the airplane and the wind.
2. (b) Draw the vector representing the airplane’s ground speed.
3. (c) Interpret the result.

(a)
(b)
(c) Answers will vary.

Air Travel An airplane maintains a constant airspeed of $500 \,\rm{km/h}$ headed due west. There is a tail wind blowing at $500 \,\rm{km/h}$ .
1. (a) Draw vectors representing the velocities of the airplane and the tail wind.
2. (b) Draw the vector representing the airplane’s ground speed.
3. (c) Interpret the result.

Suppose $\mathbf{v}$ and $\mathbf{w}$ are nonzero vectors represented by arrows with the same initial point, and that the terminal points of $\mathbf{v}$ and $\mathbf{w}$ are $P$ and $Q$ , respectively. Suppose the vector $\mathbf{u}$ is represented by an arrow from the initial point of $\mathbf{v}$ to the midpoint of the directed line segment $\overrightarrow{\it PQ}$ . Write $\mathbf{u}$ in terms of $\mathbf{v}$ and $\mathbf{w}$ .

$\mathbf{u}=\dfrac12(\mathbf{v}+\mathbf{w})$

Find nonzero scalars $a$ and $b$ so that $\begin{equation*} 4\mathbf{v}+a(\mathbf{v}-\mathbf{w})+b(\mathbf{v}+\mathbf{w})=\mathbf{0} \end{equation*}$

for every pair of vectors $\mathbf{v}$ and $\mathbf{w}$ .