Quiz: Convert vectors between magnitude-direction and i j form

📚 Quick Study Guide

• 📏 Magnitude-Direction Form: Represents a vector using its length (magnitude) and angle (direction) with respect to a reference axis (usually the positive x-axis).
• 🧭 Direction Angle: Angle $\theta$ measured counter-clockwise from the positive x-axis.
• ➕ i j Form: Represents a vector as a sum of its horizontal (i) and vertical (j) components: $\vec{v} = ai + bj$, where $a$ and $b$ are scalars.
• 🔄 Conversion Formulas:
  • Magnitude-Direction to i j: $a = r \cos(\theta)$, $b = r \sin(\theta)$, where $r$ is the magnitude and $\theta$ is the direction angle.
  • i j to Magnitude-Direction: $r = \sqrt{a^2 + b^2}$, $\theta = \arctan(\frac{b}{a})$ (adjust quadrant as needed).
• 🧭 Quadrant Adjustment: When finding the angle $\theta$ using $\arctan(\frac{b}{a})$, consider the quadrant of the vector (determined by the signs of $a$ and $b$) to ensure the angle is in the correct range (0° to 360°).

✍️ Practice Quiz

1. A vector has a magnitude of 10 and a direction of 30°. What is its i component?
   1. 5
   2. $5\sqrt{3}$
   3. 10
   4. $10\sqrt{3}$
2. A vector has a magnitude of 5 and a direction of 120°. What is its j component?
   1. 2.5
   2. $-2.5$
   3. $2.5\sqrt{3}$
   4. $-2.5\sqrt{3}$
3. A vector is given by $3i + 4j$. What is its magnitude?
   1. 5
   2. 7
   3. 25
   4. 12
4. A vector is given by $-5i + 5j$. What is its direction (in degrees)?
   1. 45
   2. 135
   3. 225
   4. 315
5. A vector has a magnitude of 7 and a direction of 270°. What is its i j form?
   1. $7i + 0j$
   2. $0i + 7j$
   3. $0i - 7j$
   4. $-7i + 0j$
6. A vector is given by $-2i - 2\sqrt{3}j$. What is its direction (in degrees)?
   1. 60
   2. 120
   3. 240
   4. 300
7. A vector has a magnitude of 12 and a direction of 0°. What is its i component?
   1. 0
   2. 6
   3. $6\sqrt{3}$
   4. 12