Calculus: Early Transcendentals

Jon Rogawski, Colin Adams, Robert Franzosa

Chapter 12

Vector Geometry - all with Video Answers

Educators

WM

Section 1

Vectors in the Plane

Problem 1

Sketch the vectors $\mathbf{v}_{1}, \mathbf{v}_{2}, \mathbf{v}_{3}, \mathbf{v}_{4}$ with tail $P$ and head $Q,$ and compute their lengths. Are any two of these vectors equivalent?

James Kiss

James Kiss

Numerade Educator

Problem 2

Sketch the vector $\mathbf{b}=\langle 3,4\rangle$ based at $P=(-2,-1)$.

Foster Wisusik

Numerade Educator

Problem 3

What is the terminal point of the vector $\mathbf{a}=\langle 1,3\rangle$ based at $P=(2,2) ?$ Sketch a and the vector $\mathrm{a}_{0}$ based at the origin and equivalent to a.

Vikash Ranjan

Numerade Educator

Problem 4

Let $\mathbf{v}=\overrightarrow{P Q},$ where $P=(1,1)$ and $Q=(2,2) .$ What is the head of the vector $v^{\prime}$ equivalent to $v$ based at (2,4) ? What is the head of the vector $\mathbf{v}_{0}$ equivalent to $\mathbf{v}$ based at the origin? Sketch $\mathbf{v}, \mathbf{v}_{0},$ and $\mathbf{v}^{\prime}$

Foster Wisusik

Numerade Educator

Problem 5

In Exercises $5-8$, refer to the unit vectors in Figure $2 I$.
Find the components of $\mathbf{u}$.

Linda Hand

Numerade Educator

Problem 6

Refer to the unit vectors in Figure $2 I$.
Find the components of $\mathbf{v}$.

Linda Hand

Numerade Educator

Problem 7

Refer to the unit vectors in Figure $2 I$.
Find the components of $\mathbf{w}$.

Linda Hand

Numerade Educator

Problem 8

Refer to the unit vectors in Figure $2 I$.
Find the components of $\mathbf{q}$.

Linda Hand

Numerade Educator

Problem 9

In Exercises 9-12, find the components of $\overrightarrow{P Q} .$
$P=(3,2), \quad Q=(2,7)$

Linda Hand

Numerade Educator

Problem 10

Find the components of $\overrightarrow{P Q} .$
$P=(-3,-5), \quad Q=(4,-6)$

Linda Hand

Numerade Educator

Problem 11

Find the components of $\overrightarrow{P Q} .$
$P=(1,-7), \quad Q=(0,17)$

Linda Hand

Numerade Educator

Problem 12

Find the components of $\overrightarrow{P Q} .$
$P=(0,2), \quad Q=(5,0)$

Linh Vu

Numerade Educator

Problem 13

In Exercises $13-20,$ calculate.
$\langle 2,1\rangle+\langle 3,4\rangle$

Linda Hand

Numerade Educator

Problem 14

Calculate.
$\langle-4,6\rangle-\langle 3,-2\rangle$

Linh Vu

Numerade Educator

Problem 15

Calculate.
$5\langle 6,2\rangle$

Linda Hand

Numerade Educator

Problem 16

Calculate.
$4(\langle 1,1\rangle+\langle 3,2\rangle)$

Linh Vu

Numerade Educator

Problem 17

Calculate.
$\left\langle-\frac{1}{2}, \frac{5}{3}\right\rangle+\left\langle 3, \frac{10}{3}\right\rangle$

Linh Vu

Numerade Educator

Problem 18

Calculate.
$2.7\langle-1.4,0.8\rangle-3.3\langle 3.1,-2.2\rangle$

Linda Hand

Numerade Educator

Problem 19

Calculate.
$(2 e, 1-2 \pi\rangle-\langle 2 e-\pi, 8-2 \pi)$

William Semus

Numerade Educator

Problem 20

Calculate.
$\left(\ln 6, \sin ^{2} 3\right)+\left\langle 1-\ln 3, \cos ^{2} 3\right\rangle$

Linda Hand

Numerade Educator

Problem 21

Which of the vectors (A)-(C) in Figure 22 is equivalent to $\mathbf{v}-\mathbf{w}$ ?

Linh Vu

Numerade Educator

Problem 22

Sketch $\mathbf{v}+\mathbf{w}$ and $\mathbf{v}-\mathbf{w}$ for the vectors in Figure 23 .

Linh Vu

Numerade Educator

Problem 23

Sketch $2 \mathbf{v},-\mathbf{w}, \mathbf{v}+\mathbf{w},$ and $2 \mathbf{v}-\mathbf{w}$ for the vectors in Figure $24 .$

Linh Vu

Numerade Educator

Problem 24

Sketch $\mathbf{v}=\langle 1,3\rangle, \mathbf{w}=\langle 2,-2\rangle, \mathbf{v}+\mathbf{w}, \mathbf{v}-\mathbf{w}$

Foster Wisusik

Numerade Educator

Problem 25

Sketch $\mathbf{v}=\langle 0,2\rangle, \mathbf{w}=\langle-2,4\rangle, 3 \mathbf{v}+\mathbf{w}, 2 \mathbf{v}-2 \mathbf{w}$.

Vikash Ranjan

Numerade Educator

Problem 26

Sketch $\mathbf{v}=\langle-2,1\rangle, \mathbf{w}=\langle 2,2\rangle, \mathbf{v}+2 \mathbf{w}, \mathbf{v}-2 \mathbf{w}$.

Foster Wisusik

Numerade Educator

Problem 27

Sketch the vector $\mathbf{v}$ such that $\mathbf{v}+\mathbf{v}_{1}+\mathbf{v}_{2}=\mathbf{0}$ for $\mathbf{v}_{1}$ and $\mathbf{v}_{2}$ in Figure $25(\mathrm{~A})$.

Linh Vu

Numerade Educator

Problem 28

Sketch the vector sum $\mathbf{v}=\mathbf{v}_{1}+\mathbf{v}_{2}+\mathbf{v}_{3}+\mathbf{v}_{4}$ in Figure $25(\mathrm{~B})$.

Linh Vu

Numerade Educator

Problem 29

Let $\mathbf{v}=\overrightarrow{P Q},$ where $P=(-2,5), Q=(1,-2)$. Which of the following vectors with the given tails and heads are equivalent to $\mathbf{v} ?$
(a) $(-3,3), \quad(0,4)$
(b) $(0,0), \quad(3,-7)$
(c) (-1,2),(2,-5)
(d) $(4,-5), \quad(1,4)$

Vikash Ranjan

Numerade Educator

Problem 30

Which of the following vectors are parallel to $\mathbf{v}=\langle 6,9\rangle$ and which point in the same direction?
(a) (12,18)
(b) (3,2)
(c) \langle 2,3\rangle
(d) (-6,-9)
(e) $\quad(-24,-27\rangle$
(f) \langle-24,-36\rangle

Foster Wisusik

Numerade Educator

Problem 31

In Exercises $31-34,$ sketch the vectors $\overrightarrow{A B}$ and $\overrightarrow{P Q},$ and determine whether they are equivalent.
$A=(1,1), \quad B=(3,7), \quad P=(4,-1), \quad Q=(6,5)$

Linda Hand

Numerade Educator

Problem 32

Sketch the vectors $\overrightarrow{A B}$ and $\overrightarrow{P Q},$ and determine whether they are equivalent.
$A=(1,4), \quad B=(-6,3), \quad P=(1,4), \quad Q=(6,3)$

Linh Vu

Numerade Educator

Problem 33

Sketch the vectors $\overrightarrow{A B}$ and $\overrightarrow{P Q},$ and determine whether they are equivalent.
$A=(-3,2), \quad B=(0,0), \quad P=(0,0), \quad Q=(3,-2)$

Linh Vu

Numerade Educator

Problem 34

Sketch the vectors $\overrightarrow{A B}$ and $\overrightarrow{P Q},$ and determine whether they are equivalent.
$A=(5,8), \quad B=(1,8), \quad P=(1,8), \quad Q=(-3,8)$

Linh Vu

Numerade Educator

Problem 35

In Exercises $35-38,$ are $\overrightarrow{A B}$ and $\overrightarrow{P Q}$ parallel? And if so, do they point in the same direction?
$A=(1,1), \quad B=(3,4), \quad P=(1,1), \quad Q=(7,10)$

Linda Hand

Numerade Educator

Problem 36

Are $\overrightarrow{A B}$ and $\overrightarrow{P Q}$ parallel? And if so, do they point in the same direction?
$A=(-3,2), \quad B=(0,0), \quad P=(0,0), \quad Q=(3,2)$

Linh Vu

Numerade Educator

Problem 37

Are $\overrightarrow{A B}$ and $\overrightarrow{P Q}$ parallel? And if so, do they point in the same direction?
$A=(2,2), \quad B=(-6,3), \quad P=(9,5), \quad Q=(17,4)$

Linh Vu

Numerade Educator

Problem 38

Are $\overrightarrow{A B}$ and $\overrightarrow{P Q}$ parallel? And if so, do they point in the same direction?
$A=(5,8), \quad B=(2,2), \quad P=(2,2), \quad Q=(-3,8)$

Linh Vu

Numerade Educator

Problem 39

In Exercises $39-42,$ let $R=(-2,7) .$ Calculate the following:
The length of $\overrightarrow{O R}$

Linda Hand

Numerade Educator

Problem 40

Let $R=(-2,7) .$ Calculate the following:
The components of $\mathbf{u}=\overrightarrow{P R},$ where $P=(1,2)$

Linh Vu

Numerade Educator

Problem 41

Let $R=(-2,7) .$ Calculate the following:
The point $P$ such that $\overrightarrow{P R}$ has components \langle-2,7\rangle

Linh Vu

Numerade Educator

Problem 42

Let $R=(-2,7) .$ Calculate the following:
The point $Q$ such that $\overrightarrow{R Q}$ has components \langle 8,-3\rangle

Linh Vu

Numerade Educator

Problem 43

In Exercises $43-52,$ find the given vector.
Unit vector $\mathrm{e}_{\mathrm{v}},$ where $\mathbf{v}=\langle 3,4\rangle$

Linda Hand

Numerade Educator

Problem 44

Find the given vector.
Unit vector $\mathrm{e}_{\mathrm{w}},$ where $\mathbf{w}=(24,7\rangle$

Linh Vu

Numerade Educator

Problem 45

Find the given vector.
Vector of length 4 in the direction of $\mathbf{u}=\langle-1,-1\rangle$

Linh Vu

Numerade Educator

Problem 46

Find the given vector.
Vector of length 3 in the direction of $\mathbf{v}=4 \mathbf{i}+3 \mathbf{j}$

Linh Vu

Numerade Educator

Problem 47

Find the given vector.
Vector of length 2 in the direction opposite to $\mathbf{v}=\mathbf{1}-\mathbf{j}$

Linda Hand

Numerade Educator

Problem 48

Find the given vector.
Unit vector in the direction opposite to $\mathbf{v}=\langle-2,4\rangle$

Linh Vu

Numerade Educator

Problem 49

Find the given vector.
Unit vector e making an angle of $\frac{4 \pi}{7}$ with the $x$ -axis

Linh Vu

Numerade Educator

Problem 50

Find the given vector.
Vector $\mathrm{v}$ of length 2 making an angle of $30^{\circ}$ with the $x$ -axis

Linh Vu

Numerade Educator

Problem 51

Find the given vector.
A unit vector pointing in the direction from (1,1) to (0,3)

Linda Hand

Numerade Educator

Problem 52

Find the given vector.
A unit vector pointing in the direction from (-3,4) to the origin

Linda Hand

Numerade Educator

Problem 53

Find all scalars $\lambda$ such that $\lambda\langle 2,3\rangle$ has length 1 .

Vikash Ranjan

Numerade Educator

Problem 54

Find a vector $\mathbf{v}$ satisfying $3 \mathbf{v}+(5,20\rangle=\langle 11,17\rangle$.

Linh Vu

Numerade Educator

Problem 55

What are the coordinates of the point $P$ in the parallelogram in Figure $26(\mathrm{~A}) ?$

Vikash Ranjan

Numerade Educator

Problem 56

What are the coordinates $a$ and $b$ in the parallelogram in Figure $26(\mathrm{~B}) ?$

Foster Wisusik

Numerade Educator

Problem 57

Let $\mathbf{v}=\overrightarrow{A B}$ and $\mathbf{w}=\overrightarrow{A C},$ where $A, B, C$ are three distinct points in the plane. Match (a) - (d) with (i) -(iv). Hint: Draw a picture.
(a) $-\mathbf{W}$
(b) $-\mathbf{v}$
(c) $\mathbf{W}-\mathbf{v}$
(d) $\mathbf{v}-\mathbf{w}$
(i) $\overrightarrow{C B}$
(ii) $\overrightarrow{C A}$
(iii) $\overrightarrow{B C}$
(iv) $\overrightarrow{B A}$

Linh Vu

Numerade Educator

Problem 58

Find the components and length of the following vectors:
(a) $4 \mathbf{i}+3 \mathbf{j}$
(b) $2 \mathbf{i}-3 j$
(c) $\mathbf{i}+\mathbf{j}$
(d) $\mathrm{i}-3 \mathrm{j}$

Foster Wisusik

Numerade Educator

Problem 59

In Exercises $59-62,$ calculate the linear combination.
$3 \mathrm{j}+(9 \mathrm{i}+4 \mathrm{j})$

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 60

Calculate the linear combination.
$-\frac{3}{2} \mathbf{i}+5\left(\frac{1}{2} \mathbf{j}-\frac{1}{2} \mathbf{i}\right)$

Linh Vu

Numerade Educator

Problem 61

Calculate the linear combination.
$(3 \mathbf{i}+\mathbf{j})-6 \mathbf{j}+2(\mathbf{j}-4 i)$

Linh Vu

Numerade Educator

Problem 62

Calculate the linear combination.
$3(3 i-4 j)+5(\mathbf{i}+4 \mathbf{j})$

Linh Vu

Numerade Educator

Problem 63

For each of the position vectors $\mathbf{u}$ with endpoints $A, B,$ and $C$ in Figure 27 , indicate with a diagram the multiples $r \mathbf{v}$ and $s \mathbf{w}$ such that $\mathbf{u}=r \mathbf{v}+s \mathbf{w} .$ A sample is shown for $\mathbf{u}=\overrightarrow{O Q}$

Vikash Ranjan

Numerade Educator

Problem 64

Sketch the parallelogram spanned by $\mathbf{v}=\langle 1,4\rangle$ and $\mathbf{w}=\langle 5,2\rangle .$ Add the vector $\mathbf{u}=\langle 2,3\rangle$ to the sketch and express $\mathbf{u}$ as a linear combination of $\mathbf{v}$ and $\mathbf{w}$.

WM

William Mead

Numerade Educator

Problem 65

In Exercises 65 and $66,$ express $\mathbf{u}$ as a linear combination $\mathbf{u}=$ $r \mathbf{v}+s \mathbf{w} .$ Then sketch $\mathbf{u}, \mathbf{v}, \mathbf{w},$ and the parallelogram formed by $r \mathbf{v}$ and $s \mathbf{w}$
$\mathbf{u}=\langle 3,-1\rangle ; \quad \mathbf{v}=(2,1\rangle, \mathbf{w}=\langle 1,3\rangle$

Linda Hand

Numerade Educator

Problem 66

Express $\mathbf{u}$ as a linear combination $\mathbf{u}=$ $r \mathbf{v}+s \mathbf{w} .$ Then sketch $\mathbf{u}, \mathbf{v}, \mathbf{w},$ and the parallelogram formed by $r \mathbf{v}$ and $s \mathbf{w}$
$\mathbf{u}=\langle 6,-2\rangle ; \quad \mathbf{v}=\langle 1,1\rangle, \mathbf{w}=\langle 1,-1\rangle$

Linda Hand

Numerade Educator

Problem 67

Calculate the magnitude of the force on cables 1 and 2 shown in Figure $28 .$

Linda Hand

Numerade Educator

Problem 68

Determine the magnitude of the forces $\mathbf{F}_{1}$ and $\mathbf{F}_{2}$ in Figure $29,$ assuming that there is no net force on the object.

Vikash Ranjan

Numerade Educator

Problem 69

A plane flying due east at $200 \mathrm{~km} / \mathrm{h}$ encounters a $40-\mathrm{km} / \mathrm{h}$ wind blowing in the northeast direction. The resultant velocity of the plane is the vector sum $\mathbf{v}=\mathbf{v}_{1}+\mathbf{v}_{2},$ where $\mathbf{v}_{1}$ is the velocity vector of the plane and $\mathbf{v}_{2}$ is the velocity vector of the wind (Figure 30 ). The angle between $\mathbf{v}_{1}$ and $\mathbf{v}_{2}$ is $\frac{\pi}{4}$. Determine the resultant speed of the plane (the length of the vector v).

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 70

In Exercises $70-72,$ refer to Figure $31,$ which shows a robotic arm consisting of two segments of lengths $L_{1}$ and $L_{2}$
Find the components of the vector $\mathbf{r}=\overrightarrow{O P}$ in terms of $\theta_{1}$ and $\theta_{2}$.

Linda Hand

Numerade Educator

Problem 71

Refer to Figure $31,$ which shows a robotic arm consisting of two segments of lengths $L_{1}$ and $L_{2}$
Let $L_{1}=5$ and $L_{2}=3$. Find $\mathbf{r}$ for $\theta_{1}=\frac{\pi}{3}, \theta_{2}=\frac{\pi}{4}$.

Linda Hand

Numerade Educator

Problem 72

Refer to Figure $31,$ which shows a robotic arm consisting of two segments of lengths $L_{1}$ and $L_{2}$
Let $L_{1}=5$ and $L_{2}=3$. Show that the set of points reachable by the robotic arm with $\theta_{1}=\theta_{2}$ is an ellipse.

Vikash Ranjan

Numerade Educator

Problem 73

Use vectors to prove that the diagonals $\overline{A C}$ and $\overline{B D}$ of a parallelogram bisect each other (Figure 32). Hint: Observe that the midpoint of $\overline{B D}$ is the terminal point of $\mathbf{w}+\frac{1}{2}(\mathbf{v}-\mathbf{w})$

Vikash Ranjan

Numerade Educator

Problem 74

Use vectors to prove that the segments joining the midpoints of opposite sides of a quadrilateral bisect each other (Figure 33). Hint: Show that the midpoints of these segments are the terminal points of $\frac{1}{4}(2 \mathrm{u}+\mathbf{v}+\mathbf{z})$ and $\frac{1}{4}(2 \mathbf{v}+\mathbf{w}+\mathbf{u})$

Vikash Ranjan

Numerade Educator

Problem 75

Prove that two nonzero vectors $\mathbf{v}=\langle a, b\rangle$ and $\mathbf{w}=\langle c, d\rangle$ are perpendicular if and only if
$$
a c+b d=0
$$

Linda Hand

Numerade Educator