1.
[latex]{\bf{w}} = 5{\bf{i}}+9{\bf{j}},[/latex] [latex]~~{\bf{v}} = 3{\bf{i}}+2{\bf{j}}[/latex]
Chapter 9: Vectors
Exercises: 9.3 The Dot Product
Skills
1. Find the component of [latex]{\bf{w}}[/latex] in the direction of [latex]{\bf{v}}[/latex] #1–6, 37–40
2. Compute the dot product #11–22, 27–36
3. Find the angle between two vectors #23–26
4. Resolve a vector into components in given directions #7–10, 41–42
Suggested Homework Problems
Problems: #2, 38, 12, 18, 22, 28, 32, 26, 8
Homework 9-3
Exercise Group
For Problems 1–6, find the component of [latex]{\bf{w}}[/latex] in the direction of [latex]{\bf{v}}\text{.}[/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]
Exercise Group
For Problems 7–10,
- Resolve [latex]{\bf{w}}[/latex] into two components, one parallel to [latex]{\bf{v}}[/latex] and the other orthogonal to [latex]{\bf{v}}\text{.}[/latex]
- Sketch both vectors and the vector components.
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]
Exercise Group
For Problems 11–18, compute the dot product [latex]{\bf{u}} \cdot {\bf{v}}\text{.}[/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.
Exercise Group
For Problems 19–22, decide whether the pair of vectors is orthogonal.
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]
Exercise Group
For Problems 23–26, find the angle between the vectors.
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]
Exercise Group
For Problems 27–30, find a value of [latex]k[/latex] so that [latex]{\bf{v}}[/latex] is orthogonal to [latex]{\bf{w}}\text{.}[/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]
Exercise Group
For Problems 31–36, evaluate the expression for the vectors
[latex]{\bf{u}}=2{\bf{i}}+5{\bf{j}},~~{\bf{v}}=-3{\bf{i}}+4{\bf{j}},~~{\bf{w}}=3{\bf{i}}-2{\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.
- 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]
- Show that [latex]{\bf{u}}[/latex] and [latex]{\bf{v}}[/latex] are orthogonal.
- 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]
- 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.
- Find unit vectors [latex]{\bf{u}}[/latex] and [latex]{\bf{v}}[/latex] in the directions of [latex]30°[/latex] and [latex]120°\text{.}[/latex]
- Show that [latex]{\bf{u}}[/latex] and [latex]{\bf{v}}[/latex] are orthogonal.
- 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]
- 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]
Exercise Group
For Problems 43–48, let [latex]{\bf{u}} = a{\bf{i}}+b{\bf{j}}[/latex] and [latex]{\bf{v}} = c{\bf{i}}+d{\bf{j}}\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.
- 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]
- Use part (a) and Problems 45 and 46 to derive the coordinate definition of [latex]{\bf{v}} \cdot {\bf{w}}\text{.}[/latex]
52.
- 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]
- Use part (a) to prove the triangle inequality:
[latex]\|{\bf{v}}+{\bf{w}}\| \le \|{\bf{v}}\|+\|{\bf{w}}\|[/latex]
53.
- 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]
- Use the figure at right to explain why part (a) proves the law of cosines.
54.
- 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]
- 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]
