Introduction
A unit vector has magnitude 1 and indicates pure direction. By combining the standard unit vectors and , we can express any 2D vector.
Prerequisite Connection
You can add vectors and perform scalar multiplication.
Today's Increment
We learn to find unit vectors, use - notation, and convert using direction angles.
Why This Matters
Unit vectors are the "building blocks" for all vectors—essential in physics, 3D graphics, and calculus.
Unit Vectors & Direction
Standard Unit Vectors
i-j Notation
Example:
Finding a Unit Vector
Divide each component by the magnitude. The "hat" notation denotes a unit vector.
Direction Angle Form
Where is the angle from the positive x-axis.
Finding the Direction Angle
Adjust for quadrant based on the signs of and .
Interactive: Unit Vector from Direction Angle
= 0.707i + 0.707j
Worked Examples
Example 1: Finding a Unit Vector
Find the unit vector in the direction of .
Step 1: Find the magnitude
Step 2: Divide each component
Solution
Example 2: Vector from Magnitude and Direction
Find the vector with magnitude 6 and direction angle .
Step 1: Find the components
Solution
Example 3: Finding Direction Angle
Find the direction angle of .
Step 1: Find reference angle
Step 2: Determine quadrant
With and , the vector is in Quadrant II.
Solution
Common Pitfalls
Forgetting to normalize
A unit vector must have magnitude 1. Always verify: .
Wrong quadrant for direction angle
only gives angles in . Check the signs of both components to find the true quadrant.
Confusing i and j
is the horizontal unit vector, is vertical. Don't swap them!
Real-World Application
Game Development: Character Movement
In video games, when a player presses "up-right," the game needs a direction vector. It's not simply because that would move faster diagonally!
Instead, games normalize to the unit vector , then multiply by the character's speed. This ensures consistent movement speed in all directions.
Practice Quiz
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