What formula does this calculator use?

It uses the dot product formula: cos(θ) = (A · B) / (|A| × |B|), where A · B is the dot product and |A|, |B| are the magnitudes. The angle θ = arccos(result) is then converted to degrees or radians.

What is the angle if both vectors are identical?

Two identical (or parallel same-direction) vectors have an angle of 0 degrees between them, since their dot product equals the product of their magnitudes.

What happens with a zero vector?

The angle is undefined when either vector is the zero vector (all components are 0), because magnitude is 0 and division by zero is not allowed. The tool will show an error in that case.

What is the maximum possible angle between two vectors?

180 degrees (π radians). This occurs when the vectors point in exactly opposite directions.

Does the magnitude of the vectors affect the angle?

No. The angle between vectors depends only on their direction, not their length. Scaling either vector by any positive constant does not change the angle.

Can I use this for 2D vectors?

Yes. Toggle to 2D mode and enter X and Y components only. The calculator uses the same dot product formula without the Z component.

Vector Angle Calculator

Enter two 2D or 3D vectors and instantly get the angle between them in degrees and radians.

Dimension Mode

A Vector A

B Vector B

Enter vector components and click
Calculate Angle

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Summary

The Vector Angle Calculator computes the angle between two vectors in 2D or 3D space using the dot product formula: cos(θ) = (A · B) / (|A| × |B|). Simply enter the X, Y (and optionally Z) components of each vector, switch between 2D and 3D mode, and get the angle in both degrees and radians instantly. This tool is useful for physics, engineering, computer graphics, and linear algebra problems.

How it works

Select 2D or 3D mode using the toggle at the top.
Enter the X and Y (and Z for 3D) components for Vector A.
Enter the X and Y (and Z for 3D) components for Vector B.
Click "Calculate Angle" or the result updates automatically.
The angle between the vectors is shown in both degrees and radians.
Use the "Swap Vectors" button to exchange the two vector inputs.

Use cases

Find the angle between force vectors in physics problems.
Compute the angle between edges or faces in 3D graphics and game development.
Verify dot product calculations in linear algebra homework.
Determine the angle of incidence between a light ray and a surface normal.
Calculate heading differences between two direction vectors in navigation.
Check perpendicularity or parallelism of vectors quickly.
Solve engineering problems involving resultant forces or velocities.
Validate rotation angles in robotics or animation rigs.

Frequently Asked Questions

Last updated: 2026-06-04 · Reviewed by Nham Vu