# 22.3: Exercises

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$

$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$

( \newcommand{\kernel}{\mathrm{null}\,}\) $$\newcommand{\range}{\mathrm{range}\,}$$

$$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$

$$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$

$$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$

$$\newcommand{\Span}{\mathrm{span}}$$

$$\newcommand{\id}{\mathrm{id}}$$

$$\newcommand{\Span}{\mathrm{span}}$$

$$\newcommand{\kernel}{\mathrm{null}\,}$$

$$\newcommand{\range}{\mathrm{range}\,}$$

$$\newcommand{\RealPart}{\mathrm{Re}}$$

$$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$

$$\newcommand{\Argument}{\mathrm{Arg}}$$

$$\newcommand{\norm}[1]{\| #1 \|}$$

$$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$

$$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\AA}{\unicode[.8,0]{x212B}}$$

$$\newcommand{\vectorA}[1]{\vec{#1}} % arrow$$

$$\newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow$$

$$\newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vectorC}[1]{\textbf{#1}}$$

$$\newcommand{\vectorD}[1]{\overrightarrow{#1}}$$

$$\newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}}$$

$$\newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}}$$

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$

## Exercise $$\PageIndex{1}$$

Graph the vectors in the plane.

1. $$\overrightarrow{PQ}$$ with $$P(2,1)$$ and $$Q(4,7)$$
2. $$\overrightarrow{PQ}$$ with $$P(-3,3)$$ and $$Q(-5,-4)$$
3. $$\overrightarrow{PQ}$$ with $$P(0,-4)$$ and $$Q(6,0)$$
4. $$\langle -2,4 \rangle$$
5. $$\langle -3,-3 \rangle$$
6. $$\langle 5,5\sqrt{2} \rangle$$

## Exercise $$\PageIndex{2}$$

Find the magnitude and directional angle of the vector.

1. $$\langle 6,8\rangle$$
2. $$\langle -2,5 \rangle$$
3. $$\langle -4,-4\rangle$$
4. $$\langle 3,-3\rangle$$
5. $$\langle 2,-2 \rangle$$
6. $$\langle 4\sqrt{3},4\rangle$$
7. $$\langle -\sqrt{3},-1\rangle$$
8. $$\langle -4,4\sqrt{3}\rangle$$
9. $$\langle -2\sqrt{3},-2\rangle$$
10. $$\overrightarrow{PQ}$$, where $$P(3,1)$$ and $$Q(7,4)$$
11. $$\overrightarrow{PQ}$$, where $$P(4,-2)$$ and $$Q(-5,7)$$
1. $$10,53^{\circ}$$
2. $$\sqrt{29}, 112^{\circ}$$
3. $$4 \sqrt{2}, 225^{\circ}$$
4. $$3 \sqrt{2},-45^{\circ}$$
5. $$2 \sqrt{2},-45^{\circ}$$
6. $$8,30^{\circ}$$
7. $$2,210^{\circ}$$
8. $$8,120^{\circ}$$
9. $$4,210^{\circ}$$
10. $$5,37^{\circ}$$
11. $$9 \sqrt{2}, 135^{\circ}$$

## Exercise $$\PageIndex{3}$$

Perform the operation on the vectors.

1. $$5\cdot \langle 3,2\rangle$$
2. $$2\cdot \langle -1,4 \rangle$$
3. $$(-10)\cdot \langle -\dfrac{3}{2},-\dfrac{7}{5} \rangle$$
4. $$\langle2,3\rangle+\langle 6,1\rangle$$
5. $$\langle 5,-4\rangle-\langle -8,-9\rangle$$
6. $$3\cdot \langle 5,3\rangle+4\cdot \langle 2,8\rangle$$
7. $$(-2) \langle -5,-4\rangle-6\langle -1,-2\rangle$$
8. $$\dfrac 2 3 \langle-3,6\rangle-\dfrac 7 5\langle10,-15\rangle$$
9. $$\sqrt{2}\cdot \langle \dfrac{\sqrt{8}}{6},\dfrac{-5\sqrt{2}}{12} \rangle-2\langle\dfrac 2 3,\dfrac 5 3\rangle$$
10. $$6\vec{i}-4\vec{j}$$
11. $$-5\vec{i}+\vec{j}+3\vec{i}$$
12. $$3\cdot\langle-4,2\rangle-8\vec{j}+12\vec{i}$$
13. Find $$4\vec{v}+7\vec{w}$$ for $$\vec{v}=\langle 2,3 \rangle$$ and $$\vec{w}=\langle 5,1\sqrt{3} \rangle$$
14. Find $$\vec{v}-2\vec{w}$$ for $$\vec{v}=\langle -11,-6 \rangle$$ and $$\vec{w}=\langle -3,2 \rangle$$
15. Find $$3\vec{v}-\vec{w}$$ for $$\vec{v}=-4\vec{i}+7\vec{j}$$ and $$\vec{w}=6\vec{i}+\vec{j}$$
16. Find $$-\vec{v}-\sqrt{5}\vec{w}$$ for $$\vec{v}=5\vec{j}$$ and $$\vec{w}=-8\vec{i}+\sqrt{5}\vec{j}$$
1. $$\langle 15,10\rangle$$
2. $$\langle-2,8\rangle$$
3. $$\langle 15,14\rangle$$
4. $$\langle 8,4\rangle$$
5. $$\langle 13,5\rangle$$
6. $$\langle 23,41\rangle$$
7. $$\langle 16,20\rangle$$
8. $$\langle-16,25\rangle$$
9. $$\left\langle-\dfrac{2}{3},-\dfrac{25}{6}\right\rangle$$
10. $$\langle 6,-4\rangle$$
11. $$\langle-2,1\rangle$$
12. $$\langle 0,-2\rangle$$
13. $$\langle 43,12+7 \sqrt{3}\rangle$$
14. $$\langle-5,-10\rangle$$
15. $$\langle-18,20\rangle$$
16. $$\langle 8 \sqrt{5},-10\rangle$$

## Exercise $$\PageIndex{4}$$

Find a unit vector in the direction of the given vector.

1. $$\langle 8,-6 \rangle$$
2. $$\langle -3,-\sqrt{7} \rangle$$
3. $$\langle 9,2 \rangle$$
4. $$\langle -\sqrt{5},\sqrt{31} \rangle$$
5. $$\langle 5\sqrt{2},3\sqrt{10} \rangle$$
6. $$\langle 0,-\dfrac{3}{5} \rangle$$
1. $$\left\langle\dfrac{4}{5},-\dfrac{3}{5}\right\rangle$$
2. $$\left\langle-\dfrac{3}{4},-\dfrac{\sqrt{7}}{4}\right\rangle$$
3. $$\left\langle\dfrac{9 \sqrt{85}}{85}, \dfrac{2 \sqrt{85}}{85}\right\rangle$$
4. $$\left\langle-\dfrac{\sqrt{5}}{6}, \dfrac{\sqrt{31}}{6}\right\rangle$$
5. $$\left\langle\dfrac{5 \sqrt{70}}{70}, \dfrac{3 \sqrt{14}}{14}\right\rangle$$
6. $$\langle 0,-1\rangle$$

## Exercise $$\PageIndex{5}$$

The vectors $$\vec{v_1}$$ and $$\vec{v_2}$$ below are being added. Find the approximate magnitude and directional angle of sum $$\vec{v}=\vec{v_1}+\vec{v_2}$$ (see Example 22.2.4).

1. $$||\vec{v_1}||=6$$, and $$\theta_1=60^\circ$$, and $$||\vec{v_2}||=2$$, and $$\theta_2=180^\circ$$
2. $$||\vec{v_1}||=3.7$$, and $$\theta_1=92^\circ$$, and $$||\vec{v_2}||=2.2$$, and $$\theta_2=253^\circ$$
3. $$||\vec{v_1}||=8$$, and $$\theta_1=\dfrac{3\pi}{4}$$, and $$||\vec{v_2}||=8\sqrt{2}$$, and $$\theta_2=\dfrac{3\pi}{2}$$
1. $$\vec{v}=\langle 1,3 \sqrt{3}\rangle,\|\vec{v}\|=2 \sqrt{7}, \theta \approx 79^{\circ}$$
2. $$\vec{v} \approx\langle-.772,1.594\rangle,\|\vec{v}\| \approx 7.63, \theta \approx 116^{\circ}$$
3. $$\vec{v}=\langle-4 \sqrt{2},-4 \sqrt{2}\rangle,\|v\|=8, \theta=225^{\circ}=\dfrac{5 \pi}{4}$$