Angle Between Vectors Calculator

How it works

1. 1Enter the components of Vector a and Vector b — switch between 2D and 3D with the dimension tabs.
2. 2The calculator computes the dot product a · b and both magnitudes |a| and |b|, then evaluates θ = acos((a · b) / (|a| |b|)).
3. 3The angle is displayed in degrees (0°–180°) and in radians so you can use the result directly in code or further calculations.

Use cases

• Computer graphics — determining the angle of incidence between a light direction and a surface normal for shading calculations.
• Physics — finding the angle between a force vector and a displacement vector to compute work done (W = F · d · cos θ).
• Machine learning — measuring the cosine similarity between two feature vectors to assess their directional relationship.

Frequently asked questions

What does an angle of 0° or 180° mean?

0° means the vectors point in exactly the same direction (they are parallel and co-directional). 180° means they point in exactly opposite directions. 90° means they are perpendicular — their dot product is zero.

Why does the calculator return — for a zero vector?

A zero vector has no direction, so the angle between it and any other vector is undefined. The formula requires dividing by the magnitudes |a| and |b|, and |0| = 0 makes that undefined.

Does the order of the vectors matter?

No. The angle between a and b equals the angle between b and a because the dot product is commutative: a · b = b · a. The result is always the same positive angle in [0°, 180°].