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

Question

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! 🤓

kimberly.peters · Accepted Answer

📚 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:
        
            Calculate the magnitude of the vector.
            Divide each component of the original vector by its magnitude.

🧭Direction: The unit vector retains the original vector's direction.

Practice Quiz

Question 1: What is the unit vector of $\vec{v} = \langle 3, 4 \rangle$?
        
            $\langle 0.6, 0.8 \rangle$
            $\langle 3, 4 \rangle$
            $\langle 0.8, 0.6 \rangle$
            $\langle 5, 5 \rangle$

Question 2: What is the magnitude of any unit vector?
        
            0
            -1
            1
            Undefined

Question 3: Find the unit vector of $\vec{v} = \langle -5, 0 \rangle$.
        
            $\langle 1, 0 \rangle$
            $\langle 0, -1 \rangle$
            $\langle -1, 0 \rangle$
            $\langle 0, 1 \rangle$

Question 4: Determine the unit vector for $\vec{v} = \langle 0, -2 \rangle$.
        
            $\langle 1, 0 \rangle$
            $\langle 0, -1 \rangle$
            $\langle -1, 0 \rangle$
            $\langle 0, 1 \rangle$

Question 5: What is the unit vector of $\vec{v} = \langle 1, 1, 1 \rangle$?
        
            $\langle \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}} \rangle$
            $\langle 1, 1, 1 \rangle$
            $\langle \frac{1}{3}, \frac{1}{3}, \frac{1}{3} \rangle$
            $\langle 0, 0, 0 \rangle$

Question 6: What is the process of finding the unit vector called?
        
            Normalization
            Vectorization
            Magnification
            Differentiation

Question 7: Calculate the unit vector of $\vec{v} = \langle 2, -1, 3 \rangle$.
        
            $\langle \frac{2}{\sqrt{14}}, \frac{-1}{\sqrt{14}}, \frac{3}{\sqrt{14}} \rangle$
            $\langle \frac{2}{\sqrt{4}}, \frac{-1}{\sqrt{1}}, \frac{3}{\sqrt{9}} \rangle$
            $\langle 2, -1, 3 \rangle$
            $\langle 0, 0, 0 \rangle$

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

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📚 Quick Study Guide

Practice Quiz

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