Position Vector Test Questions for High School Pre-Calculus Exams

📚 Quick Study Guide

• 📍 Definition: A position vector describes the location of a point in space relative to the origin.
• 🧭 Representation: A position vector $\vec{r}$ from the origin to a point P(x, y) is represented as $\vec{r} = \langle x, y \rangle$ in 2D and $\vec{r} = \langle x, y, z \rangle$ in 3D.
• ➕ Vector Addition: To add position vectors $\vec{a} = \langle x_1, y_1 \rangle$ and $\vec{b} = \langle x_2, y_2 \rangle$, simply add their corresponding components: $\vec{a} + \vec{b} = \langle x_1 + x_2, y_1 + y_2 \rangle$.
• ➖ Vector Subtraction: Similarly, subtract corresponding components: $\vec{a} - \vec{b} = \langle x_1 - x_2, y_1 - y_2 \rangle$.
• 🔢 Scalar Multiplication: Multiplying a position vector $\vec{a} = \langle x, y \rangle$ by a scalar $k$ gives $k\vec{a} = \langle kx, ky \rangle$.
• 📏 Magnitude: The magnitude (or length) of a position vector $\vec{r} = \langle x, y \rangle$ is calculated as $|\vec{r}| = \sqrt{x^2 + y^2}$ in 2D and $|\vec{r}| = \sqrt{x^2 + y^2 + z^2}$ in 3D.
• 🎯 Unit Vector: A unit vector in the direction of $\vec{r}$ is found by dividing $\vec{r}$ by its magnitude: $\hat{r} = \frac{\vec{r}}{|\vec{r}|}$.

🧪 Practice Quiz

1. What is the position vector of a point located at coordinates (3, -4)?
   1. $\langle -3, 4 \rangle$
   2. $\langle 3, 4 \rangle$
   3. $\langle 3, -4 \rangle$
   4. $\langle -4, 3 \rangle$
2. If $\vec{a} = \langle 2, 5 \rangle$ and $\vec{b} = \langle -1, 3 \rangle$, what is $\vec{a} + \vec{b}$?
   1. $\langle 3, 8 \rangle$
   2. $\langle 1, 8 \rangle$
   3. $\langle 1, 2 \rangle$
   4. $\langle -2, 15 \rangle$
3. What is the magnitude of the position vector $\vec{r} = \langle 5, -12 \rangle$?
   1. 7
   2. 13
   3. 17
   4. $\sqrt{119}$
4. Given a position vector $\vec{v} = \langle 4, -3 \rangle$, find a unit vector in the same direction.
   1. $\langle \frac{4}{7}, -\frac{3}{7} \rangle$
   2. $\langle \frac{4}{5}, -\frac{3}{5} \rangle$
   3. $\langle 1, -1 \rangle$
   4. $\langle -4, 3 \rangle$
5. A point is located 6 units along the x-axis and 8 units along the y-axis. What is its position vector?
   1. $\langle 8, 6 \rangle$
   2. $\langle -6, -8 \rangle$
   3. $\langle 6, 8 \rangle$
   4. $\langle -8, -6 \rangle$
6. If $\vec{p} = \langle 1, -2, 3 \rangle$ and $\vec{q} = \langle -2, 4, -1 \rangle$, what is $\vec{p} - \vec{q}$?
   1. $\langle -1, 2, 2 \rangle$
   2. $\langle 3, -6, 4 \rangle$
   3. $\langle -3, 6, -4 \rangle$
   4. $\langle 3, 2, -2 \rangle$
7. What is $3\vec{a}$ if $\vec{a} = \langle -2, 1 \rangle$?
   1. $\langle -6, 3 \rangle$
   2. $\langle -2, 3 \rangle$
   3. $\langle 6, -3 \rangle$
   4. $\langle -6, 1 \rangle$