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Unit Vector

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Unit Vector

Study Guide

Key Definition

A unit vector is a vector of magnitude $1$.

Important Notes

A unit vector has a magnitude of $1$.
Unit vectors are often used to indicate direction.
They can be represented in different coordinate systems.
Unit vectors help in simplifying vector calculations.
Common unit vectors are $\hat{i}$, $\hat{j}$, $\hat{k}$ in 3D space.

Mathematical Notation

$\hat{v}$ denotes a unit vector.

$|\vec{v}|$ represents the magnitude of vector $\vec{v}$.

$\sqrt{x^2 + y^2}$ calculates the magnitude in 2D space.

$\sqrt{x^2 + y^2 + z^2}$ calculates the magnitude in 3D space.

Remember to use proper notation when solving problems

Why It Works

Unit vectors allow for easy representation of direction in vector space, simplifying many mathematical and physical problems.

Remember

The magnitude of any unit vector is always $1$.

Quick Reference

Magnitude Formula:$\sqrt{x^2 + y^2}$

3D Magnitude Formula:$\sqrt{x^2 + y^2 + z^2}$

Understanding Unit Vector

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Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

A unit vector has a magnitude of $1$ and is used to indicate direction.

Practice Problems

Test your understanding with practice problems

Real-World Problem

Question Exercise

Intermediate

Skateboarder Displacement

A skateboarder moves from point P = (2, 3) to Q = (5, 7). Find the unit vector representing the direction of motion.

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

If $\vec{a} = (3, 4)$, find the unit vector in the direction of $\vec{a}$.

Challenge Quiz

Single Choice Quiz

Advanced

Find the unit vector of $\vec{b} = (6, 8, 10)$.

Recap

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Review key concepts and takeaways