- Demonstrate the ability to work with a linear combination of unit vectors
- Demonstrate the ability to find the dot product of two vectors
- Demonstrate the ability to find the angle between two vectors
- Demonstrate the ability to determine if two vectors are orthogonal
#1:
Instructions: Find the unit vector u in the direction of v.
$$a)\hspace{.1em}\overrightarrow{v}=-4\hat{\text{i}}+ 8\hat{\text{j}}$$
$$b)\hspace{.1em}\overrightarrow{v}=\langle 30, 40 \rangle$$
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#2:
Instructions: Find the dot product of the vectors u and v.
$$a)\hspace{.1em}$$ $$\overrightarrow{u}=\langle -8, -3 \rangle$$ $$ \overrightarrow{v}=\langle 6, 6 \rangle$$
$$b)\hspace{.1em}$$ $$\overrightarrow{u}=-9\hat{\text{i}}- 2\hat{\text{j}}$$ $$\overrightarrow{v}=-3\hat{\text{i}}- 8\hat{\text{j}}$$
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#3:
Instructions: Find the measure of the angle θ between vectors u and v geometrically.
$$a)\hspace{.1em}$$ $$\overrightarrow{u}=\langle 8, 3 \rangle$$ $$\overrightarrow{v}=\langle 2, 7 \rangle$$
Hint: Sketch the vectors u and v in standard position. Form a triangle using u - v. Use the law of cosines to find the angle between u and v.
Instructions: Find the measure of the angle θ between vectors u and v.
$$b)\hspace{.1em}$$ $$\overrightarrow{u}=\langle 2, -4 \rangle$$ $$ \overrightarrow{v}=\langle 4, -5 \rangle$$
$$c)\hspace{.1em}$$ $$\overrightarrow{u}=\langle 2, 9 \rangle$$ $$ \overrightarrow{v}=\langle 8, -3 \rangle$$
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#4:
Instructions: Find the measure of the angle θ between vectors u and v.
$$a)\hspace{.1em}$$ $$\overrightarrow{u}=-\hat{\text{i}}- 3\hat{\text{j}}$$ $$\overrightarrow{v}=3\hat{\text{i}}+ 8\hat{\text{j}}$$
$$b)\hspace{.1em}$$ $$\overrightarrow{u}=-3\hat{\text{i}}+ 5\hat{\text{j}}$$ $$\overrightarrow{v}=3\hat{\text{i}}- 8\hat{\text{j}}$$
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#5:
Instructions: Determine if the vectors u and v are orthogonal.
$$a)\hspace{.1em}$$ $$\overrightarrow{u}=\langle -5, -2 \rangle$$ $$ \overrightarrow{v}=\langle -2, -5 \rangle$$
$$b)\hspace{.1em}$$ $$\overrightarrow{u}=9\hat{\text{i}}- 4\hat{\text{j}}$$ $$\overrightarrow{v}=-8\hat{\text{i}}- 18\hat{\text{j}}$$
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Written Solutions:
#1:
Solutions:
$$a)\hspace{.1em}\hat{\text{u}}=-\frac{\sqrt{5}}{5}\hat{\text{i}}+ \frac{2 \sqrt{5}}{5}\hat{\text{j}}$$
$$b)\hspace{.1em}\hat{\text{u}}=\left\langle \frac{3}{5}, \frac{4}{5}\right\rangle$$
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#2:
Solutions:
$$a)\hspace{.1em}\overrightarrow{u}\cdot \overrightarrow{v}=-66$$
$$b)\hspace{.1em}\overrightarrow{u}\cdot \overrightarrow{v}=43$$
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#3:
Solutions:
$$a)$$ 
$$\left| \hspace{.2em}\overrightarrow{u}- \overrightarrow{v}\hspace{.2em}\right|=2\sqrt{13}$$ $$\left| \hspace{.2em}\overrightarrow{u}\hspace{.2em}\right|=\sqrt{73}$$ $$\left| \hspace{.2em}\overrightarrow{v}\hspace{.2em}\right|=\sqrt{53}$$ Law of Cosines: $$\left| \hspace{.2em}\overrightarrow{u}- \overrightarrow{v}\hspace{.2em}\right|^2=\left| \hspace{.2em}\overrightarrow{u}\hspace{.2em}\right|^2 + \left| \hspace{.2em}\overrightarrow{v}\hspace{.2em}\right|^2 - 2\left| \hspace{.2em}\overrightarrow{u}\hspace{.2em}\right| \left| \hspace{.2em}\overrightarrow{v}\hspace{.2em}\right| \cdot \text{cos}\hspace{.2em}θ \hspace{.2em}$$ $$(2\sqrt{13})^2=(\sqrt{73})^2 + (\sqrt{53})^2 - 2(\sqrt{73})(\sqrt{53}) \cdot \text{cos}\hspace{.2em}θ \hspace{.2em}$$ $$52=73 + 53 - 2(\sqrt{73})(\sqrt{53}) \cdot \text{cos}\hspace{.2em}θ \hspace{.2em}$$ Solve for cos θ: $$\text{cos}\hspace{.2em}θ=\frac{-74}{- 2(\sqrt{73})(\sqrt{53})}$$ $$\text{cos}\hspace{.2em}θ=\frac{37}{\sqrt{73}\cdot \sqrt{53}}$$ $$θ ≈ 53.5°$$
$$b)\hspace{.1em}θ ≈ 12.09°$$
$$c)\hspace{.1em}θ ≈ 98.03°$$
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#4:
Solutions:
$$a)\hspace{.1em}θ ≈ 177.88°$$
$$b)\hspace{.1em}θ ≈ 169.59°$$
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#5:
Solutions:
$$a)\hspace{.1em}\text{Not Orthogonal}$$
$$b)\hspace{.1em}\text{Orthogonal}$$
