Pre-Calculus Test Questions: Calculating Angles with the Dot Product

📚 Quick Study Guide

• 📐 The dot product of two vectors, $\mathbf{a}$ and $\mathbf{b}$, is defined as: $\mathbf{a} \cdot \mathbf{b} = |\mathbf{a}| |\mathbf{b}| \cos(\theta)$, where $\theta$ is the angle between the vectors.
• 🧮 The magnitude of a vector $\mathbf{a} = (a_1, a_2)$ is calculated as: $|\mathbf{a}| = \sqrt{a_1^2 + a_2^2}$. For a 3D vector $\mathbf{a} = (a_1, a_2, a_3)$, it's $|\mathbf{a}| = \sqrt{a_1^2 + a_2^2 + a_3^2}$.
• 💡 To find the angle $\theta$ between two vectors using the dot product, rearrange the formula: $\cos(\theta) = \frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{a}| |\mathbf{b}|}$, and then $\theta = \arccos(\frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{a}| |\mathbf{b}|})$.
• 📝 Remember that the dot product can also be calculated as: $\mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2$ (in 2D) or $\mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 + a_3b_3$ (in 3D).
• ✨ The angle $\theta$ is typically given in radians or degrees. Make sure your calculator is in the correct mode!

Practice Quiz

1. Question 1: What is the dot product of vectors $\mathbf{a} = (2, 3)$ and $\mathbf{b} = (1, -1)$?
   1. -1
   2. 1
   3. 5
   4. -5
2. Question 2: Find the angle between vectors $\mathbf{a} = (1, 0)$ and $\mathbf{b} = (0, 1)$.
   1. 0
   2. $\frac{\pi}{2}$
   3. $\pi$
   4. $\frac{3\pi}{2}$
3. Question 3: Calculate the cosine of the angle between $\mathbf{a} = (1, 2)$ and $\mathbf{b} = (3, 4)$.
   1. $\frac{11}{5\sqrt{2}}$
   2. $\frac{11}{5}$
   3. $\frac{5\sqrt{2}}{11}$
   4. $\frac{1}{2}$
4. Question 4: If $\mathbf{a} \cdot \mathbf{b} = 0$, what can you conclude about the angle between $\mathbf{a}$ and $\mathbf{b}$?
   1. The vectors are parallel.
   2. The vectors are perpendicular.
   3. The angle is acute.
   4. The angle is obtuse.
5. Question 5: Determine the angle between $\mathbf{a} = (2, 2, 0)$ and $\mathbf{b} = (0, 2, 2)$.
   1. $\frac{\pi}{6}$
   2. $\frac{\pi}{4}$
   3. $\frac{\pi}{3}$
   4. $\frac{\pi}{2}$
6. Question 6: What is the magnitude of the vector $\mathbf{v} = (3, -4)$?
   1. 5
   2. 7
   3. 25
   4. $\sqrt{7}$
7. Question 7: Given vectors $\mathbf{u} = (5, -2)$ and $\mathbf{v} = (-1, 3)$, find the value of $\mathbf{u} \cdot \mathbf{v}$.
   1. -11
   2. 1
   3. 11
   4. -1