22.3: Exercises

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. $\langle 5,5\sqrt{2} \rangle$

Answer

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 $Q(-5,7)$

Answer

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 $\vec{w}=-8\vec{i}+\sqrt{5}\vec{j}$

Answer

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. $\langle 0,-\dfrac{3}{5} \rangle$

Answer

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 $\theta_2=\dfrac{3\pi}{2}$

Answer

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. $\vec{v}=\langle-4 \sqrt{2},-4 \sqrt{2}\rangle,\|v\|=8, \theta=225^{\circ}=\dfrac{5 \pi}{4}$