🧠Direction of the Vector Calculator
Find the direction and angles of a 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.
Click Calculate
Press "Calculate Direction" to find the unit vector, direction cosines, and angles.
View Results
See the unit vector (direction), direction cosines, angles, and direction ratios displayed.
Formula
Unit Vector = v / |v| = (x/|v|, y/|v|, z/|v|)
|v| = √(x² + y² + z²)
Direction Cosines: cos(α) = x/|v|, cos(β) = y/|v|, cos(γ) = z/|v|
Key Formulas:
- Magnitude: |v| = √(x² + y² + z²)
- Unit vector: û = v / |v| = (x/|v|, y/|v|, z/|v|)
- Direction cosines: cos(α), cos(β), cos(γ) are the unit vector components
- Angle in XY-plane: θ = arctan2(y, x)
- Angle from z-axis: φ = arccos(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
Angle: θ = arctan2(4, 3) ≈ 53.13°
About Direction of the Vector Calculator
The Direction of the Vector Calculator finds the direction of a vector by calculating its unit vector (normalized vector), direction cosines, and angles. The direction is independent of the vector's magnitude.
When to Use This Calculator
- Physics: Find direction of velocity, force, or acceleration vectors
- Engineering: Analyze direction of forces, moments, and displacements
- Mathematics: Normalize vectors and find direction cosines
- 3D Graphics: Calculate directions for rendering and transformations
- Robotics: Determine orientation and direction of movement
- Vector Analysis: Understand vector directions and relationships
Why Use Our Calculator?
- ✅ Unit Vector: Normalizes the vector to show pure direction
- ✅ Direction Cosines: Shows cosines of angles with axes
- ✅ Angle Calculations: Computes angles in XY-plane and from z-axis
- ✅ 2D and 3D Support: Works with 2D and 3D vectors
- ✅ Direction Ratios: Shows simple integer ratios when possible
- ✅ Completely Free: No registration required
Understanding Vector Direction
The direction of a vector is represented by its unit vector (vector divided by its magnitude). Direction cosines are the components of the unit vector, representing the cosines of angles the vector makes with the coordinate axes.
- Unit vector: Vector divided by its magnitude (length = 1)
- Direction cosines: cos(α), cos(β), cos(γ) are the unit vector components
- Sum of squares: cos²(α) + cos²(β) + cos²(γ) = 1
- Angle in XY-plane: arctan2(y, x) gives angle from positive x-axis
- Angle from z-axis: arccos(z/|v|) for 3D vectors
- Direction ratios: Simplest integer representation of direction
Real-World Applications
Physics: When analyzing forces, the direction of a force vector determines its effect. Unit vectors help separate magnitude from direction.
Navigation: Direction vectors represent headings and bearings. Normalizing gives compass directions independent of speed.
Computer Graphics: Light directions, camera orientations, and surface normals are represented as unit vectors in 3D rendering.
Frequently Asked Questions
What is the direction of a vector?
The direction of a vector is represented by its unit vector (vector divided by magnitude). It shows which way the vector points, independent of its length. Formula: û = v / |v|.
What are direction cosines?
Direction cosines are the components of the unit vector: cos(α) = x/|v|, cos(β) = y/|v|, cos(γ) = z/|v|. They represent the cosines of angles the vector makes with x, y, and z axes.
What is a unit vector?
A unit vector has magnitude 1 and points in the same direction as the original vector. It's obtained by dividing the vector by its magnitude: û = v / |v|.
Can a zero vector have direction?
No! A zero vector (0, 0, 0) has no direction because it has zero magnitude. You cannot divide by zero to find a unit vector.
What are direction ratios?
Direction ratios are the simplest integer representation of a vector's direction, obtained by dividing components by their greatest common divisor. For example, (6, 8, 10) has direction ratios 3:4:5.
How do direction cosines relate to angles?
Direction cosines are cosines of angles: α = angle with x-axis, β = angle with y-axis, γ = angle with z-axis. Their sum of squares equals 1: cos²(α) + cos²(β) + cos²(γ) = 1.
What's the difference between direction and magnitude?
Magnitude is the length/size of the vector. Direction is which way it points. Unit vectors represent pure direction (magnitude = 1), while original vectors include both magnitude and direction.