🧠Unit Vector Calculator
Calculate unit vector (normalized vector)
Vector Components
How to Use This Calculator
Enter Vector Components
Input the x, y, and optionally z components of your vector. Leave z blank (or 0) for 2D vectors.
Calculate
Press "Calculate Unit Vector" to find the normalized vector (unit vector) with magnitude 1.
View Results
See the unit vector, direction cosines, and angles displayed. The unit vector magnitude is 1.
Formula
Unit Vector: û = v / |v|
where |v| = √(x² + y² + z²)
Direction cosines: cos(α) = x/|v|, cos(β) = y/|v|, cos(γ) = z/|v|
Example: Vector v = (3, 4, 0)
Magnitude: |v| = √(3² + 4²) = √25 = 5
Unit vector: û = (3/5, 4/5, 0) = (0.6, 0.8, 0)
Direction cosines: cos(α) = 0.6, cos(β) = 0.8, cos(γ) = 0
About Unit Vector Calculator
The Unit Vector Calculator finds the unit vector (normalized vector) of any vector. A unit vector has magnitude 1 and points in the same direction as the original vector.
When to Use This Calculator
- Physics: Normalize velocity, force, or acceleration vectors
- Engineering: Find direction vectors for forces and moments
- Mathematics: Normalize vectors in vector operations
- 3D Graphics: Calculate normalized vectors for lighting and rendering
- Robotics: Determine normalized direction vectors
Understanding Unit Vectors
A unit vector has magnitude 1 and represents pure direction. It's obtained by dividing a vector by its magnitude: û = v / |v|. Unit vectors are used to represent directions without magnitude.
Frequently Asked Questions
What is a unit vector?
A unit vector has magnitude 1 and points in the same direction as the original vector. Formula: û = v / |v|, where |v| is the magnitude of vector v.
Why normalize vectors?
Unit vectors represent pure direction without magnitude. They're useful when direction matters but magnitude doesn't, such as in lighting calculations, force directions, and velocity directions.
What are direction cosines?
Direction cosines are the components of the unit vector: cos(α), cos(β), cos(γ) where α, β, γ are angles the vector makes with x, y, z axes. They sum to 1: cos²(α) + cos²(β) + cos²(γ) = 1.
Can a zero vector have a unit vector?
No! A zero vector (0, 0, 0) has magnitude 0, so you cannot divide by zero to get a unit vector. Zero vectors have no direction.
How do I check if a vector is already a unit vector?
Calculate its magnitude. If |v| = 1 (or very close to 1), it's already a unit vector. If |v| ≠1, normalize it by dividing by its magnitude.