Test Objectives
  • 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
Dot Product & Angle Between Vectors Practice Test:

#1:

Instructions: Find the unit vector u in the direction of v.

$$a)\hspace{.1em}\overrightarrow{v}=-4\hat{\text{i}}+ 8\hat{\text{j}}$$

$$b)\hspace{.1em}\overrightarrow{v}=\langle 30, 40 \rangle$$


#2:

Instructions: Find the dot product of the vectors u and v.

$$a)\hspace{.1em}$$ $$\overrightarrow{u}=\langle -8, -3 \rangle$$ $$ \overrightarrow{v}=\langle 6, 6 \rangle$$

$$b)\hspace{.1em}$$ $$\overrightarrow{u}=-9\hat{\text{i}}- 2\hat{\text{j}}$$ $$\overrightarrow{v}=-3\hat{\text{i}}- 8\hat{\text{j}}$$


#3:

Instructions: Find the measure of the angle θ between vectors u and v.

$$a)\hspace{.1em}$$ $$\overrightarrow{u}=\langle 2, -4 \rangle$$ $$ \overrightarrow{v}=\langle 4, -5 \rangle$$

$$b)\hspace{.1em}$$ $$\overrightarrow{u}=\langle 2, 9 \rangle$$ $$ \overrightarrow{v}=\langle 8, -3 \rangle$$


#4:

Instructions: Find the measure of the angle θ between vectors u and v.

$$a)\hspace{.1em}$$ $$\overrightarrow{u}=-\hat{\text{i}}- 3\hat{\text{j}}$$ $$\overrightarrow{v}=3\hat{\text{i}}+ 8\hat{\text{j}}$$

$$b)\hspace{.1em}$$ $$\overrightarrow{u}=-3\hat{\text{i}}+ 5\hat{\text{j}}$$ $$\overrightarrow{v}=3\hat{\text{i}}- 8\hat{\text{j}}$$


#5:

Instructions: Determine if the vectors u and v are orthogonal.

$$a)\hspace{.1em}$$ $$\overrightarrow{u}=\langle -5, -2 \rangle$$ $$ \overrightarrow{v}=\langle -2, -5 \rangle$$

$$b)\hspace{.1em}$$ $$\overrightarrow{u}=9\hat{\text{i}}- 4\hat{\text{j}}$$ $$\overrightarrow{v}=-8\hat{\text{i}}- 18\hat{\text{j}}$$


Written Solutions:

#1:

Solutions:

$$a)\hspace{.1em}\hat{\text{u}}=-\frac{\sqrt{5}}{5}\hat{\text{i}}+ \frac{2 \sqrt{5}}{5}\hat{\text{j}}$$

$$b)\hspace{.1em}\hat{\text{u}}=\left\langle \frac{3}{5}, \frac{4}{5}\right\rangle$$


#2:

Solutions:

$$a)\hspace{.1em}\overrightarrow{u}\cdot \overrightarrow{v}=-66$$

$$b)\hspace{.1em}\overrightarrow{u}\cdot \overrightarrow{v}=43$$


#3:

Solutions:

$$a)\hspace{.1em}θ ≈ 12.09°$$

$$b)\hspace{.1em}θ ≈ 98.03°$$


#4:

Solutions:

$$a)\hspace{.1em}θ ≈ 177.88°$$

$$b)\hspace{.1em}θ ≈ 169.59°$$


#5:

Solutions:

$$a)\hspace{.1em}\text{Not Orthogonal}$$

$$b)\hspace{.1em}\text{Orthogonal}$$