victoria455
victoria455 14h ago • 0 views

Solved Examples: Finding Unit Vectors in 2D and 3D Space

Hey there! 👋 Ever get stuck trying to figure out unit vectors? Don't worry, it's easier than it looks! This guide breaks down the concept and gives you a quiz to test your understanding. Let's get started! 🤓
🧮 Mathematics

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📚 Quick Study Guide

  • 📏Definition: A unit vector is a vector with a magnitude (or length) of 1. It points in the same direction as the original vector.
  • 🔢Formula for 2D: Given a vector $\vec{v} = \langle a, b \rangle$, the unit vector $\hat{u}$ is calculated as $\hat{u} = \frac{\vec{v}}{\|\vec{v}\|} = \frac{\langle a, b \rangle}{\sqrt{a^2 + b^2}}$.
  • Formula for 3D: Given a vector $\vec{v} = \langle a, b, c \rangle$, the unit vector $\hat{u}$ is calculated as $\hat{u} = \frac{\vec{v}}{\|\vec{v}\|} = \frac{\langle a, b, c \rangle}{\sqrt{a^2 + b^2 + c^2}}$.
  • 💡Key Steps:
    1. Calculate the magnitude of the vector.
    2. Divide each component of the original vector by its magnitude.
  • 🧭Direction: The unit vector retains the original vector's direction.

Practice Quiz

  1. Question 1: What is the unit vector of $\vec{v} = \langle 3, 4 \rangle$?
    1. $\langle 0.6, 0.8 \rangle$
    2. $\langle 3, 4 \rangle$
    3. $\langle 0.8, 0.6 \rangle$
    4. $\langle 5, 5 \rangle$
  2. Question 2: What is the magnitude of any unit vector?
    1. 0
    2. -1
    3. 1
    4. Undefined
  3. Question 3: Find the unit vector of $\vec{v} = \langle -5, 0 \rangle$.
    1. $\langle 1, 0 \rangle$
    2. $\langle 0, -1 \rangle$
    3. $\langle -1, 0 \rangle$
    4. $\langle 0, 1 \rangle$
  4. Question 4: Determine the unit vector for $\vec{v} = \langle 0, -2 \rangle$.
    1. $\langle 1, 0 \rangle$
    2. $\langle 0, -1 \rangle$
    3. $\langle -1, 0 \rangle$
    4. $\langle 0, 1 \rangle$
  5. Question 5: What is the unit vector of $\vec{v} = \langle 1, 1, 1 \rangle$?
    1. $\langle \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}} \rangle$
    2. $\langle 1, 1, 1 \rangle$
    3. $\langle \frac{1}{3}, \frac{1}{3}, \frac{1}{3} \rangle$
    4. $\langle 0, 0, 0 \rangle$
  6. Question 6: What is the process of finding the unit vector called?
    1. Normalization
    2. Vectorization
    3. Magnification
    4. Differentiation
  7. Question 7: Calculate the unit vector of $\vec{v} = \langle 2, -1, 3 \rangle$.
    1. $\langle \frac{2}{\sqrt{14}}, \frac{-1}{\sqrt{14}}, \frac{3}{\sqrt{14}} \rangle$
    2. $\langle \frac{2}{\sqrt{4}}, \frac{-1}{\sqrt{1}}, \frac{3}{\sqrt{9}} \rangle$
    3. $\langle 2, -1, 3 \rangle$
    4. $\langle 0, 0, 0 \rangle$
Click to see Answers
  1. A
  2. C
  3. C
  4. B
  5. A
  6. A
  7. A

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