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11.8: Vectors

Last updated

Mar 4, 2023
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- 11.7: More Functions and Identities
- 11.9: Polar Coordinates and Complex Numbers

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9.1 Geometric Form

Homework 9-1

1. $Screen Shot 2023-02-27 at 2.42.01 AM.png$

3. $Screen Shot 2023-02-27 at 2.42.13 AM.png$

5. $Screen Shot 2023-02-27 at 2.42.21 AM.png$

7. A and E

9. H and K

11. $Screen Shot 2023-02-27 at 2.43.33 AM.png$

13. $Screen Shot 2023-02-27 at 2.43.42 AM.png$

15. $Screen Shot 2023-02-27 at 2.43.55 AM.png$

17. $Screen Shot 2023-02-27 at 2.44.03 AM.png$

19. $Screen Shot 2023-02-27 at 5.49.47 PM.png$ $\|\mathbf{A}\|=\sqrt{13}, \theta=-33.7^{\circ}$

21. $Screen Shot 2023-02-27 at 5.49.55 PM.png$ $\|\mathbf{C}\|=1, \quad \theta=90^{\circ}$

23. $Screen Shot 2023-02-27 at 5.50.03 PM.png$ $\| \mathbf{E} \| = 5, \quad \theta = 90^{\circ}$

25. $Screen Shot 2023-02-27 at 5.50.16 PM.png$ $\| \mathbf{G} = 4, \quad \theta = 180^{\circ}$

27. $\|\mathbf{v}\|=13, \quad \theta=-67.38^{\circ}$

29. $\|\mathbf{v}\|=\sqrt{85} \approx 9.22, \theta=229.4^{\circ}$

31. $Screen Shot 2023-02-27 at 5.50.31 PM.png$ $\| v + w\| = 32.9, \quad \theta = 109.3^{\circ}$

33. $Screen Shot 2023-02-27 at 5.56.24 PM.png$ $\|\mathbf{v}+\mathbf{w}\|=11.4, \theta=162.4^{\circ}$

35. $Screen Shot 2023-02-27 at 5.56.33 PM.png$ $4.47 \mathrm{mi}, 23.4^{\circ}$ east of north

37. $Screen Shot 2023-02-27 at 5.56.48 PM.png$ $129.4 \mathrm{mph}, 85.4^{\circ}$ west of north

39.

a $v_x=10, v_y=10 \sqrt{3}, w_x=5 \sqrt{2}, w_y=-5 \sqrt{2}$

b $19.9 \mathrm{mph}, 59^{\circ}$ east of north

41.

a $v_x \approx-1.23, v_y \approx 3.38, w_x \approx-0.32, w_y \approx-2.23$

b $1.9 \mathrm{~km}, 54.5^{\circ}$ west of north

43. $Screen Shot 2023-02-27 at 5.59.10 PM.png$

45. $Screen Shot 2023-02-27 at 5.59.19 PM.png$

47. $Screen Shot 2023-02-27 at 5.59.29 PM.png$

49. $Screen Shot 2023-02-27 at 5.59.36 PM.png$

51. $u_x=2, u_y=1, v_x=1, v_y=-3, A_x=1, A_y=4 ; A_x=u_x-v_x, A_y=u_y-v_y$

9.2 Coordinate Form

Homework 9-2

1. $\mathbf{u}=3 \mathbf{i}+2 \mathbf{j}$

a $\sqrt{13}$

b $6 \mathbf{i}+4 \mathbf{j}$

c $2 \sqrt{13}$

3. $\mathbf{w}=6 \mathbf{i}-3 \mathbf{j}$

a $3 \sqrt{5}$

b $-6 \mathbf{i}+3 \mathbf{j}$

c $3 \sqrt{5}$

a $\mathbf{u}+\mathbf{v}=-2 \mathbf{i}+5 \mathbf{j}$ and $\|\mathbf{u}+\mathbf{v}\|=\sqrt{29}$

b $\|\mathbf{u}\|+\|\mathbf{v}\| \geq\|\mathbf{u}+\mathbf{v}\|$

a $Screen Shot 2023-02-27 at 6.05.03 PM.png$ $-5 \mathbf{i}+8 \mathbf{j}$

b $\|\mathbf{v}\|=\sqrt{89}, \quad \theta=122^{\circ}$

a $Screen Shot 2023-02-27 at 6.05.10 PM.png$ $-2 \mathbf{i}-\mathbf{j}$

b $\|\mathbf{v}\|=\sqrt{5}, \quad \theta=206.6^{\circ}$

11.

a $18 \mathbf{i}+12 \mathbf{j}$

b $\|\mathbf{v}\|=6 \sqrt{13}, \quad \theta=33.7^{\circ}$

13. $\|\mathbf{v}\|=6 \sqrt{2}, \quad \theta=135^{\circ}$

15. $\|\mathbf{w}\|=14, \quad \theta=-30^{\circ}$

17. $\|\mathbf{q}\|=4 \sqrt{745}, \quad \theta=61.56^{\circ}$

19. $\mathbf{v}=3 \sqrt{2} \mathbf{i}-3 \sqrt{2} \mathbf{j}$

21. $\mathbf{v} \approx 6.629 \mathbf{i}+4.995 \mathbf{j}$

23. $\mathbf{i} - 2\mathbf{j}$

$Screen Shot 2023-02-27 at 6.11.52 PM.png$

25. $-4 \mathbf{i} + 4 \mathbf{j}$

$Screen Shot 2023-02-27 at 6.12.00 PM.png$

27. $12 \mathbf{i}+3 \mathbf{j}$

29. $2.8 \mathbf{i}+1.9 \mathbf{j}$

31. $-3 \mathbf{i}+7 \mathbf{j}$

33. $-8 \mathbf{i}-20 \mathbf{j}$

35. $14 \mathbf{i}-9 \mathbf{j}$

37. $-9 \mathbf{i}+23 \mathbf{j}$

39. $\dfrac{-12}{13} \mathbf{i}+\dfrac{5}{13} \mathbf{j}$

41. $\dfrac{1}{\sqrt{2}} \mathbf{i}-\dfrac{1}{\sqrt{2}} \mathbf{j}$

43. $24 \mathbf{i}+45 \mathbf{j}$

45. $\dfrac{-12}{\sqrt{10}} \mathbf{i}+\dfrac{4}{\sqrt{10}} \mathbf{j}$

47.

a $Screen Shot 2023-02-27 at 6.15.28 PM.png$

b $\mathbf{u}=2.393 \mathbf{i}+1.016 \mathbf{j}, \quad \mathbf{v}=-4.242 \mathbf{i}-3.956 \mathbf{j}$

c $-1.849 \mathbf{i}-2.940 \mathbf{j}$

49.

a $Screen Shot 2023-02-27 at 6.16.09 PM.png$

b $\mathbf{u}=-11.97 \mathbf{i}+32.889 \mathbf{j}, \quad \mathbf{v}=-57.955 \mathbf{i}+15.529 \mathbf{j}$

c $45.98 \mathbf{i}+17.36 \mathbf{j}$

51.

a $Screen Shot 2023-02-27 at 6.28.05 PM.png$

b $1700 \mathrm{~m}, 28.1^{\circ}$ east of south

53.

a $Screen Shot 2023-02-27 at 6.28.49 PM.png$

b $21.98 \mathrm{~km}, 2.27^{\circ}$ north of west

55.

a $Screen Shot 2023-02-27 at 6.29.29 PM.png$

b $83 \mathrm{mi}, 62^{\circ}$ east of north

57.

a $-4 \mathbf{i}-5 \mathbf{j}$

b $4 \mathbf{i}+5 \mathbf{j}$

59.

a $\mathbf{i}-3 \mathbf{j}$

b $-\mathbf{i}+3 \mathbf{j}$

61.

a $\|\mathbf{v}\|=10,2\|\mathbf{v}\|=20=2 \cdot 10$

b $\|k \mathbf{v}\|=\sqrt{(k a)^2+(k b)^2}=k \sqrt{a^2+b^2}$

9.3 The Dot Product

Homework 9-3

1. $\dfrac{33}{\sqrt{13}}$

3. $\dfrac{-1}{\sqrt{2}}$

5. $-2\sqrt{5}$

a $\mathbf{w}=\left(\dfrac{56}{13} \mathbf{i}+\dfrac{84}{13} \mathbf{j}\right)+\left(\dfrac{48}{13} \mathbf{i}-\dfrac{32}{13} \mathbf{j}\right)$

b $Screen Shot 2023-02-27 at 8.12.46 PM.png$

a $\mathbf{w}=(4 \mathbf{i}-4 \mathbf{j})+(2 \mathbf{i}+2 \mathbf{j})$

b $Screen Shot 2023-02-27 at 8.12.54 PM.png$

11. 22

13. 0

15. 12

17. -318.2

19. not orthogonal

21. orthogonal

23. $4.4^{\circ}$

25. $97.1^{\circ}$

27. 8

29. -10

31. -21

33. $42 \mathbf{i}-28 \mathbf{j}$

35. 4

37. 38.57 lbs

39. 1289 lbs

41.

a $\dfrac{1}{\sqrt{2}} \mathbf{i}+\dfrac{1}{\sqrt{2}} \mathbf{j}$ and $\dfrac{-1}{\sqrt{2}} \mathbf{i}+\dfrac{1}{\sqrt{2}} \mathbf{j}$

b $\mathbf{u} \cdot \mathbf{v}=0$

c $\dfrac{11}{\sqrt{2}}$ and $\dfrac{5}{\sqrt{2}}$

d $Screen Shot 2023-02-27 at 8.16.16 PM.png$

43. $\mathbf{v} \cdot \mathbf{v}=c^2+d^2$

45. $k \mathbf{u} \cdot \mathbf{v}=k a c+k b d=k(a c+b d)=(a k c+b k d)$

47.

$\begin{aligned} (\mathbf{u}-\mathbf{v}) \cdot(\mathbf{u}+\mathbf{v}) & =(a-c)(a+c)+(b-d)(b+d) \\ & =\left(a^2+b^2\right)-\left(c^2+d^2\right) \end{aligned}$

49. $\dfrac{a \cdot 1+b \cdot 0}{1}=a \text { and } \dfrac{a \cdot 0+b \cdot 1}{1}=b$

51.

a Both $\mathbf{i} \cdot \mathbf{i}=1$ and $\mathbf{j} \cdot \mathbf{j}=1$ because $1 \cdot 1 \cos 0=1 ; \mathbf{i} \cdot \mathbf{j}=1 \cdot 1 \cos 90^{\circ}=0$

b $(a \mathbf{i}+b \mathbf{j}) \cdot(c \mathbf{i}+d \mathbf{j})=a c(1)+a d(0)+b c(0)+b d(1)=a c+b d$

53.

a $\|\mathbf{u}-\mathbf{v}\|^2=\mathbf{u} \cdot \mathbf{u}-2 \mathbf{u} \cdot \mathbf{v}+\mathbf{v} \cdot \mathbf{v}=\|\mathbf{u}\|^2+\|\mathbf{v}\|^2-2\|\mathbf{u}\|\|\mathbf{v}\| \cos \theta$

b Let $a=\|\mathbf{u}\|, b=\|\mathbf{v}\|, c=\|\mathbf{u}-\mathbf{v}\|$ , and $C=\theta$

9.4 Chapter Summary and Review

Review Problems

1. $Screen Shot 2023-02-27 at 8.19.44 PM.png$ $v_N=8.45 \mathrm{mph}, v_E=-18.13 \mathrm{mph}$

3. $Screen Shot 2023-02-27 at 8.20.17 PM.png$ $v_N=-1127.63 \mathrm{lbs}, v_E=-410.42 \mathrm{lbs}$

5. $\|\mathbf{A}\|=10.9, \theta=236.3^{\circ}$

7. $\mathbf{i}-\sqrt{3} \mathbf{j}$

a $Screen Shot 2023-02-27 at 8.22.32 PM.png$ $15 \mathbf{i}+3 \mathbf{j}$

b $\|\mathbf{v}\|=15.3, \theta=11.3^{\circ}$

11.

a $Screen Shot 2023-02-27 at 8.22.42 PM.png$ $2 \mathbf{i} - 6 \mathbf{j}$

b $\|\mathbf{v}\|=6.3 \mathrm{mi}, \theta=288.4^{\circ}$

13.

a $Screen Shot 2023-02-27 at 8.22.54 PM.png$

b $7.64 \mathrm{~km}, \theta=30.31^{\circ}$

15.

a $Screen Shot 2023-02-27 at 8.23.03 PM.png$

b $8.46 \mathrm{mi}, \theta=155.6^{\circ}$

17.

a $\mathbf{F}_1 = -200\mathbf{i}, \quad \mathbf{F}_2 = -60\sqrt{2}\mathbf{i} - 60\sqrt{2}\mathbf{j}, \quad \mathbf{F}_3 = 50\sqrt{3}\mathbf{i} + 50\mathbf{j}, \quad \mathbf{F}_4 = -125\mathbf{i} + 125\sqrt{3}\mathbf{j}$

b $-73.25 \mathbf{i}+181.65 \mathbf{j}$

19. $13 \mathbf{i}+5 \mathbf{j}$

21. $-7 \mathbf{i}-14 \mathbf{j}$

23. $\dfrac{2}{\sqrt{13}} \mathbf{i}+\dfrac{3}{\sqrt{13}} \mathbf{j}$

25. $\dfrac{-6}{\sqrt{29}} \mathrm{i}-\dfrac{15}{\sqrt{29}} \mathrm{j}$

27. −3.45

29. −8.08

31. $106.26^{\circ}$