1.
[latex]{\bf{w}} = 5{\bf{i}}+9{\bf{j}},[/latex] [latex]~~{\bf{v}} = 3{\bf{i}}+2{\bf{j}}[/latex]
Chapter 9: Vectors
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
For Problems 1–6, find the component of [latex]{\bf{w}}[/latex] in the direction of [latex]{\bf{v}}\text{.}[/latex]
[latex]{\bf{w}} = 5{\bf{i}}+9{\bf{j}},[/latex] [latex]~~{\bf{v}} = 3{\bf{i}}+2{\bf{j}}[/latex]
[latex]{\bf{w}} = 7{\bf{i}}+4{\bf{j}},[/latex] [latex]~~{\bf{v}} = 2{\bf{i}}+3{\bf{j}}[/latex]
[latex]{\bf{w}} = -6{\bf{i}}+5{\bf{j}},[/latex] [latex]~~{\bf{v}} = {\bf{i}}+{\bf{j}}[/latex]
[latex]{\bf{w}} = 10{\bf{i}}-14{\bf{j}},~~{\bf{v}} = {\bf{i}}+{\bf{j}}[/latex]
[latex]{\bf{w}} = 4{\bf{i}}-3{\bf{j}},[/latex] [latex]~~{\bf{v}} = -{\bf{i}}+2{\bf{j}}[/latex]
[latex]{\bf{w}} = -2{\bf{i}}-3{\bf{j}},~~{\bf{v}} = {\bf{i}}-2{\bf{j}}[/latex]
For Problems 7–10,
[latex]{\bf{w}} = 8{\bf{i}}+4{\bf{j}},[/latex] [latex]~~{\bf{v}} = 2{\bf{i}}+3{\bf{j}}[/latex]
[latex]{\bf{w}} = -3{\bf{i}}+7{\bf{j}},[/latex] [latex]~~{\bf{v}} = 4{\bf{i}}+2{\bf{j}}[/latex]
[latex]{\bf{w}} = 6{\bf{i}}-2{\bf{j}},[/latex] [latex]~~{\bf{v}} = {\bf{i}}-{\bf{j}}[/latex]
[latex]{\bf{w}} = -5{\bf{i}}+3{\bf{j}},[/latex] [latex]~~{\bf{v}} = -{\bf{i}}-3{\bf{j}}[/latex]
For Problems 11–18, compute the dot product [latex]{\bf{u}} \cdot {\bf{v}}\text{.}[/latex]
[latex]{\bf{u}} = 3{\bf{i}}+7{\bf{j}},[/latex] [latex]~~{\bf{v}} = -2{\bf{i}}+4{\bf{j}}[/latex]
[latex]{\bf{u}} = -1.3{\bf{i}}+5.6{\bf{j}},[/latex] [latex]~~{\bf{v}} = 3{\bf{i}}-5{\bf{j}}[/latex]
[latex]{\bf{u}} = 3{\bf{i}}-4{\bf{j}},[/latex] [latex]~~{\bf{v}} = 20{\bf{i}}+15{\bf{j}}[/latex]
[latex]{\bf{u}} = 2{\bf{i}}+{\bf{j}},[/latex] [latex]~~{\bf{v}} = 6{\bf{i}}+3{\bf{j}}[/latex]
[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]
[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]
For Problems 19–22, decide whether the pair of vectors is orthogonal.
[latex]2{\bf{i}}+3{\bf{j}}~[/latex] and [latex]-3{\bf{i}}-2{\bf{j}}[/latex]
[latex]-5{\bf{i}}+7{\bf{j}}~[/latex] and [latex]~7{\bf{i}}+5{\bf{j}}[/latex]
[latex]4{\bf{i}}+6{\bf{j}}~[/latex] and [latex]-15{\bf{i}}+10{\bf{j}}[/latex]
[latex]3{\bf{i}}-4{\bf{j}}~[/latex] and [latex]-3{\bf{i}}+4{\bf{j}}[/latex]
For Problems 23–26, find the angle between the vectors.
[latex]3{\bf{i}}+5{\bf{j}}~[/latex] and [latex]~2{\bf{i}}+4{\bf{j}}[/latex]
[latex]{\bf{i}}-2{\bf{j}}~[/latex] and [latex]-2{\bf{i}}-3{\bf{j}}[/latex]
[latex]4{\bf{i}}-8{\bf{j}}~[/latex] and [latex]~6{\bf{i}}+4{\bf{j}}[/latex]
[latex]-6{\bf{i}}+8{\bf{j}}~[/latex] and [latex]~18{\bf{i}}-24{\bf{j}}[/latex]
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]
[latex]{\bf{w}}=8{\bf{i}}-3{\bf{j}},[/latex] [latex]~{\bf{v}}= 3{\bf{i}}+k{\bf{j}}[/latex]
[latex]{\bf{w}}=2{\bf{i}}+7{\bf{j}},[/latex] [latex]~{\bf{v}}=k{\bf{i}}+4{\bf{j}}[/latex]
[latex]{\bf{w}}=-2{\bf{i}}-5{\bf{j}},[/latex] [latex]~{\bf{v}}=k{\bf{i}}+4{\bf{j}}[/latex]
[latex]{\bf{w}}=5{\bf{i}}+3{\bf{j}},[/latex] [latex]~{\bf{v}}=-2{\bf{i}}+k{\bf{j}}[/latex]
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]
[latex]{\bf{w}} \cdot ({\bf{u}}+{\bf{v}})[/latex]
[latex]{\bf{w}} \cdot {\bf{u}} + {\bf{w}} \cdot {\bf{v}}[/latex]
[latex]({\bf{u}} \cdot {\bf{v}}) {\bf{w}}[/latex]
[latex]({\bf{u}} \cdot {\bf{v}})({\bf{u}} \cdot {\bf{w}})[/latex]
[latex]({\bf{u}}+{\bf{v}}) \cdot ({\bf{u}}-{\bf{v}})[/latex]
[latex]\dfrac{{\bf{w}} \cdot {\bf{v}}}{{\bf{w}} \cdot {\bf{w}}} {\bf{w}}[/latex]
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.
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.
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.
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.
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]
Show that [latex]{\bf{v}} \cdot {\bf{v}} = \|{\bf{v}}\|^2\text{.}[/latex]
If [latex]\|{\bf{u}}\|=1\text{,}[/latex] show that [latex]\text{comp}_{\bf{u}}{\bf{v}}={\bf{u}} \cdot {\bf{v}}\text{.}[/latex]
Show that [latex]k{\bf{u}} \cdot {\bf{v}} = k({\bf{u}} \cdot {\bf{v}}) = {\bf{u}} \cdot k{\bf{v}}\text{.}[/latex]
Prove the distributive law: [latex]{\bf{u}} \cdot ({\bf{v}}+{\bf{w}})= {\bf{u}} \cdot {\bf{v}}+{\bf{u}} \cdot {\bf{w}}\text{.}[/latex]
Show that [latex]({\bf{u}}-{\bf{v}}) \cdot ({\bf{u}}+{\bf{v}}) = \|{\bf{u}}\|^2 - \|{\bf{v}}\|^2\text{.}[/latex]
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]
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]
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]