Exercises: 9.3 The Dot Product

1.

[latex]{\bf{w}} = 5{\bf{i}}+9{\bf{j}},[/latex] [latex]~~{\bf{v}} = 3{\bf{i}}+2{\bf{j}}[/latex]

2.

[latex]{\bf{w}} = 7{\bf{i}}+4{\bf{j}},[/latex] [latex]~~{\bf{v}} = 2{\bf{i}}+3{\bf{j}}[/latex]

3.

[latex]{\bf{w}} = -6{\bf{i}}+5{\bf{j}},[/latex] [latex]~~{\bf{v}} = {\bf{i}}+{\bf{j}}[/latex]

4.

[latex]{\bf{w}} = 10{\bf{i}}-14{\bf{j}},~~{\bf{v}} = {\bf{i}}+{\bf{j}}[/latex]

5.

[latex]{\bf{w}} = 4{\bf{i}}-3{\bf{j}},[/latex] [latex]~~{\bf{v}} = -{\bf{i}}+2{\bf{j}}[/latex]

6.

[latex]{\bf{w}} = -2{\bf{i}}-3{\bf{j}},~~{\bf{v}} = {\bf{i}}-2{\bf{j}}[/latex]

7.

[latex]{\bf{w}} = 8{\bf{i}}+4{\bf{j}},[/latex] [latex]~~{\bf{v}} = 2{\bf{i}}+3{\bf{j}}[/latex]

8.

[latex]{\bf{w}} = -3{\bf{i}}+7{\bf{j}},[/latex] [latex]~~{\bf{v}} = 4{\bf{i}}+2{\bf{j}}[/latex]

9.

[latex]{\bf{w}} = 6{\bf{i}}-2{\bf{j}},[/latex] [latex]~~{\bf{v}} = {\bf{i}}-{\bf{j}}[/latex]

10.

[latex]{\bf{w}} = -5{\bf{i}}+3{\bf{j}},[/latex] [latex]~~{\bf{v}} = -{\bf{i}}-3{\bf{j}}[/latex]

11.

[latex]{\bf{u}} = 3{\bf{i}}+7{\bf{j}},[/latex] [latex]~~{\bf{v}} = -2{\bf{i}}+4{\bf{j}}[/latex]

12.

[latex]{\bf{u}} = -1.3{\bf{i}}+5.6{\bf{j}},[/latex] [latex]~~{\bf{v}} = 3{\bf{i}}-5{\bf{j}}[/latex]

13.

[latex]{\bf{u}} = 3{\bf{i}}-4{\bf{j}},[/latex] [latex]~~{\bf{v}} = 20{\bf{i}}+15{\bf{j}}[/latex]

14.

[latex]{\bf{u}} = 2{\bf{i}}+{\bf{j}},[/latex] [latex]~~{\bf{v}} = 6{\bf{i}}+3{\bf{j}}[/latex]

15.

[latex]{\bf{u}}[/latex] has magnitude 3 and direction [latex]27°\text{,}[/latex] and [latex]{\bf{u}}[/latex] has magnitude 8 and direction [latex]33°\text{.}[/latex]

16.

[latex]{\bf{u}}[/latex] has magnitude [latex]\sqrt{7}[/latex] and direction [latex]112°\text{,}[/latex] and [latex]{\bf{u}}[/latex] has magnitude [latex]\sqrt{14}[/latex] and direction [latex]157°\text{.}[/latex]

17.

18.

19.

[latex]2{\bf{i}}+3{\bf{j}}~[/latex] and [latex]-3{\bf{i}}-2{\bf{j}}[/latex]

20.

[latex]-5{\bf{i}}+7{\bf{j}}~[/latex] and [latex]~7{\bf{i}}+5{\bf{j}}[/latex]

21.

[latex]4{\bf{i}}+6{\bf{j}}~[/latex] and [latex]-15{\bf{i}}+10{\bf{j}}[/latex]

22.

[latex]3{\bf{i}}-4{\bf{j}}~[/latex] and [latex]-3{\bf{i}}+4{\bf{j}}[/latex]

23.

[latex]3{\bf{i}}+5{\bf{j}}~[/latex] and [latex]~2{\bf{i}}+4{\bf{j}}[/latex]

24.

[latex]{\bf{i}}-2{\bf{j}}~[/latex] and [latex]-2{\bf{i}}-3{\bf{j}}[/latex]

25.

[latex]4{\bf{i}}-8{\bf{j}}~[/latex] and [latex]~6{\bf{i}}+4{\bf{j}}[/latex]

26.

[latex]-6{\bf{i}}+8{\bf{j}}~[/latex] and [latex]~18{\bf{i}}-24{\bf{j}}[/latex]

27.

[latex]{\bf{w}}=8{\bf{i}}-3{\bf{j}},[/latex] [latex]~{\bf{v}}= 3{\bf{i}}+k{\bf{j}}[/latex]

28.

[latex]{\bf{w}}=2{\bf{i}}+7{\bf{j}},[/latex] [latex]~{\bf{v}}=k{\bf{i}}+4{\bf{j}}[/latex]

29.

[latex]{\bf{w}}=-2{\bf{i}}-5{\bf{j}},[/latex] [latex]~{\bf{v}}=k{\bf{i}}+4{\bf{j}}[/latex]

30.

[latex]{\bf{w}}=5{\bf{i}}+3{\bf{j}},[/latex] [latex]~{\bf{v}}=-2{\bf{i}}+k{\bf{j}}[/latex]

31.

[latex]{\bf{w}} \cdot ({\bf{u}}+{\bf{v}})[/latex]

32.

[latex]{\bf{w}} \cdot {\bf{u}} + {\bf{w}} \cdot {\bf{v}}[/latex]

33.

[latex]({\bf{u}} \cdot {\bf{v}}) {\bf{w}}[/latex]

34.

[latex]({\bf{u}} \cdot {\bf{v}})({\bf{u}} \cdot {\bf{w}})[/latex]

35.

[latex]({\bf{u}}+{\bf{v}}) \cdot ({\bf{u}}-{\bf{v}})[/latex]

36.

[latex]\dfrac{{\bf{w}} \cdot {\bf{v}}}{{\bf{w}} \cdot {\bf{w}}} {\bf{w}}[/latex]

37.

Gary pulls a loaded wagon along a flat road. The handle of the wagon makes an angle of [latex]50°[/latex] to the horizontal. If Gary pulls with a force of 60 pounds, find the component of the force in the direction of motion.

38.

Wassily is trying to topple a statue by pulling on a rope tied to the statue’s upraised arm. The rope is making a [latex]35°[/latex] angle from horizontal. If Wassily is pulling on the rope with a force of 250 pounds, find the component of the force in the horizontal direction.

39.

An SUV weighing 6200 pounds is parked on a hill with slope [latex]12°\text{.}[/latex] Find the force needed to keep the SUV from rolling down the hill.

40.

Steve’s boat is headed due north, and the sail points at an angle of [latex]15°[/latex] east of north. The wind is blowing in the direction [latex]60°[/latex] west of south, but because of the difference in air pressure between the front and back surfaces of the sail, the boat experiences a force of 400 pounds in the direction the sail is facing. Find the component of the force in the direction of the boat’s motion.

41.

1. Find unit vectors [latex]{\bf{u}}[/latex] and [latex]{\bf{v}}[/latex] in the directions of [latex]{\bf{i+j}}[/latex] and [latex]{\bf{i-j}}\text{.}[/latex]
2. Show that [latex]{\bf{u}}[/latex] and [latex]{\bf{v}}[/latex] are orthogonal.
3. Find the components of [latex]{\bf{w}} = 3{\bf{i}}+8{\bf{j}}[/latex] in the directions of [latex]{\bf{u}}[/latex] and [latex]{\bf{v}}\text{.}[/latex]
4. Sketch the vectors [latex]{\bf{u}},~{\bf{v}}[/latex] and [latex]{\bf{w}}\text{,}[/latex] and show the components of [latex]{\bf{w}}\text{.}[/latex]

42.

1. Find unit vectors [latex]{\bf{u}}[/latex] and [latex]{\bf{v}}[/latex] in the directions of [latex]30°[/latex] and [latex]120°\text{.}[/latex]
2. Show that [latex]{\bf{u}}[/latex] and [latex]{\bf{v}}[/latex] are orthogonal.
3. Find the components of [latex]{\bf{w}} = -4{\bf{i}}+4{\bf{j}}[/latex] in the directions of [latex]{\bf{u}}[/latex] and [latex]{\bf{v}}\text{.}[/latex]
4. Sketch the vectors [latex]{\bf{u}},~{\bf{v}}[/latex] and [latex]{\bf{w}}\text{,}[/latex] and show the components of [latex]{\bf{w}}\text{.}[/latex]

43.

Show that [latex]{\bf{v}} \cdot {\bf{v}} = \|{\bf{v}}\|^2\text{.}[/latex]

44.

If [latex]\|{\bf{u}}\|=1\text{,}[/latex] show that [latex]\text{comp}_{\bf{u}}{\bf{v}}={\bf{u}} \cdot {\bf{v}}\text{.}[/latex]

45.

Show that [latex]k{\bf{u}} \cdot {\bf{v}} = k({\bf{u}} \cdot {\bf{v}}) = {\bf{u}} \cdot k{\bf{v}}\text{.}[/latex]

46.

Prove the distributive law: [latex]{\bf{u}} \cdot ({\bf{v}}+{\bf{w}})= {\bf{u}} \cdot {\bf{v}}+{\bf{u}} \cdot {\bf{w}}\text{.}[/latex]

47.

Show that [latex]({\bf{u}}-{\bf{v}}) \cdot ({\bf{u}}+{\bf{v}}) = \|{\bf{u}}\|^2 - \|{\bf{v}}\|^2\text{.}[/latex]

48.

If [latex]\|{\bf{u}}\|=\|{\bf{v}}\|\text{,}[/latex] show that [latex]{\bf{u}}+{\bf{v}}[/latex] is perpendicular to [latex]{\bf{u}}-{\bf{v}}\text{.}[/latex]

49.

Show that the component of [latex]{\bf{v}} = a{\bf{i}}+b{\bf{j}}[/latex] in the direction of [latex]{\bf{i}}[/latex] is [latex]a\text{,}[/latex] and the component of [latex]{\bf{v}}[/latex] in the direction of [latex]{\bf{j}}[/latex] is [latex]b\text{.}[/latex]

50.

Show that the dot product [latex]{\bf{u}} \cdot {\bf{v}}[/latex] gives the length of [latex]{\bf{v}}[/latex] times the component of [latex]{\bf{u}}[/latex] in the direction of [latex]{\bf{v}}\text{.}[/latex]

51.

1. Start from the geometric definition [latex]{\bf{v}} \cdot {\bf{w}} = \|{\bf{v}}\|\|{\bf{w}}\| \cos \theta[/latex] and show that [latex]{\bf{i}} \cdot {\bf{i}}=1,~~{\bf{j}} \cdot {\bf{j}}=1[/latex] and [latex]{\bf{i}} \cdot {\bf{j}}=0\text{.}[/latex]
2. Use part (a) and Problems 45 and 46 to derive the coordinate definition of [latex]{\bf{v}} \cdot {\bf{w}}\text{.}[/latex]

52.

1. Show that [latex]({\bf{v}}+{\bf{w}}) \cdot ({\bf{v}}+{\bf{w}}) = \|{\bf{v}}\|^2 + \|{\bf{w}}\|^2 + 2({\bf{v}} \cdot {\bf{w}})[/latex]
2. Use part (a) to prove the triangle inequality:
   [latex]\|{\bf{v}}+{\bf{w}}\| \le \|{\bf{v}}\|+\|{\bf{w}}\|[/latex]

53.

1. Use the dot product to show that [latex]\|{\bf{u}}-{\bf{v}}\|^2 = \|{\bf{u}}\|^2+\|{\bf{v}}\|^2 - 2\|{\bf{u}}\|\|{\bf{v}}\|\cos \theta\text{.}[/latex]
2. Use the figure at right to explain why part (a) proves the law of cosines.

54.

1. If [latex]{\bf{u}}[/latex] is a unit vector, and the angle between [latex]{\bf{u}}[/latex] and [latex]{\bf{i}}[/latex] is [latex]\alpha\text{,}[/latex] show that [latex]{\bf{u}} = \cos \alpha {\bf{i}} + \sin \alpha {\bf{j}}\text{.}[/latex]
2. Suppose [latex]{\bf{u}}[/latex] and [latex]{\bf{v}}[/latex] are unit vectors, as shown in the figure at right. Use the dot product [latex]{\bf{u}} \cdot {\bf{v}}[/latex] to prove that[latex]\cos (\beta - \alpha) = \cos \beta \cos \alpha + \sin \beta \sin \alpha[/latex]