GEOMETRYCoordinate GeometryMathematics Calculator
→

Vector Magnitude and Direction

A vector has magnitude (length) and direction. Compute |v| = √(x²+y²+z²), unit vector v̂ = v/|v|, and direction angle θ = arctan(y/x) for 2D.

Concept Fundamentals
|v| = √(x²+y²+z²)
Magnitude
v̂ = v/|v|
Unit vector

Did our AI summary help? Let us know.

î=(1,0,0), ĵ=(0,1,0), k̂=(0,0,1) are standard basis vectors. Compare squared magnitudes to avoid square roots. Zero vector (0,0,0) has undefined direction.

Key quantities
|v| = √(x²+y²+z²)
Magnitude
Key relation
v̂ = v/|v|
Unit vector
Key relation

Ready to run the numbers?

Why: Vectors represent force, velocity, acceleration — quantities with both size and direction.

How: Magnitude is the length: |v| = √(x²+y²+z²). Unit vector has magnitude 1: v̂ = v/|v|. In 2D, direction angle θ = arctan(y/x).

î=(1,0,0), ĵ=(0,1,0), k̂=(0,0,1) are standard basis vectors.Compare squared magnitudes to avoid square roots.
Sources:Wikipedia - Euclidean vector

Run the calculator when you are ready.

Start CalculatingEnter vector components (x, y) or (x, y, z).

Enter Vector Components

For educational and informational purposes only. Verify with a qualified professional.

🧮 Fascinating Math Facts

📐

Vectors represent force, velocity, acceleration.

— Physics

💻

3D positions, normals, and velocities are all vectors.

— Graphics

Key Takeaways

  • • A vector has both magnitude and direction; it is represented by components (x,y)(x, y) or (x,y,z)(x, y, z).
  • • The magnitude is ∣vecv∣=sqrtx2+y2+z2|\\vec{v}| = \\sqrt{x^2 + y^2 + z^2} — the length of the vector.
  • • The unit vector hatv=vecv/∣vecv∣\\hat{v} = \\vec{v}/|\\vec{v}| has magnitude 1 and points in the same direction.
  • • In 2D, the direction angle from the positive x-axis is theta=arctan(y/x)\\theta = \\arctan(y/x).
  • • The zero vector (0, 0, 0) has magnitude 0; its direction and unit vector are undefined.

Did You Know?

Physics

Vectors represent force, velocity, acceleration — quantities with both size and direction.

Computer Graphics

3D positions, normals, and velocities are all vectors. Unit vectors are used for lighting.

Navigation

GPS uses 3D vectors to represent your position and velocity relative to Earth.

Machine Learning

Feature vectors in ML are n-dimensional; magnitude and direction matter for similarity.

Engineering

Structural forces and moments are vector quantities analyzed by component and magnitude.

Unit Vectors

î = (1,0,0), ĵ = (0,1,0), k̂ = (0,0,1) are the standard basis vectors in 3D.

Understanding Vectors

A vector in 2D or 3D space is defined by its components. The magnitude (length) is:

∣v⃗∣=x2+y2+z2|\vec{v}| = \sqrt{x^2 + y^2 + z^2}

The unit vector (direction only) is:

v^=v⃗∣v⃗∣=(x∣v⃗∣,y∣v⃗∣,z∣v⃗∣)\hat{v} = \frac{\vec{v}}{|\vec{v}|} = \left(\frac{x}{|\vec{v}|}, \frac{y}{|\vec{v}|}, \frac{z}{|\vec{v}|}\right)

Expert Tips

Avoid Square Roots

When comparing magnitudes, compare squared magnitudes: if |v|² < |w|² then |v| < |w|.

Normalization

Always check |v| > 0 before computing the unit vector to avoid division by zero.

2D vs 3D

For 2D vectors, use z=0. The direction angle is only defined in 2D.

Component Form

Component form (x, y, z) is equivalent to xî + yĵ + zk̂ in standard basis.

Frequently Asked Questions

What is a vector?

A vector is a quantity with both magnitude (length) and direction. It is represented by components (x, y) in 2D or (x, y, z) in 3D.

How do I find the magnitude?

Use |v| = √(x² + y² + z²). For 2D, set z=0 so |v| = √(x² + y²).

What is a unit vector?

A unit vector has magnitude 1. Compute it by dividing the vector by its magnitude: v̂ = v/|v|.

When is the direction angle defined?

Only for 2D vectors. It is the angle from the positive x-axis: θ = arctan(y/x).

What about the zero vector?

The zero vector (0,0,0) has magnitude 0. Its direction and unit vector are undefined.

Can I use negative components?

Yes. Negative components point in the opposite direction along that axis.

What is component form?

Component form is the ordered tuple (x, y, z) that specifies the vector's projection onto each axis.

How to Use This Calculator

  1. Enter the vector components (x, y) for 2D or (x, y, z) for 3D. Toggle "3D mode" if needed.
  2. Click "Calculate" to compute magnitude, unit vector, direction angle (2D), and component form.
  3. Review the results in the gradient card and metrics grid.
  4. Check the visualization to see the vector in 3D space.
  5. Copy results for homework or reports.

Disclaimer: This calculator uses standard floating-point arithmetic. For the zero vector, unit vector and direction angle are undefined. Results are suitable for educational and professional use.

👈 START HERE
⬅️Jump in and explore the concept!
AI

Related Calculators

Direction Of Vector Calculator

Direction Of Vector Calculator - Calculate and learn about coordinate-geometry concepts

Mathematics

Unit Vector Calculator

Unit Vector Calculator - Calculate and learn about coordinate-geometry concepts

Mathematics

Centroid Calculator

Centroid Calculator - Calculate and learn about coordinate-geometry concepts

Mathematics

Endpoint Calculator

Endpoint Calculator - Calculate and learn about coordinate-geometry concepts

Mathematics

Gradient Calculator

Gradient Calculator - Calculate and learn about coordinate-geometry concepts

Mathematics

Rotation Calculator

Rotation Calculator - Calculate and learn about coordinate-geometry concepts

Mathematics