Angle Between Two Vectors Calculator

About this calculator

The angle between two vectors comes straight from the dot product formula: cos θ = (a · b) / (|a| |b|), where a · b is the sum of the products of matching components and |a|, |b| are the vectors' magnitudes. Taking the arccosine of that ratio gives θ directly, in radians, which this calculator also converts to degrees.

This is one of the most-used relationships in 3D math: computer graphics and game engines use it for lighting calculations and collision detection, physicists use it to resolve forces and torques, and robotics engineers use it for joint angles and motion planning. Because the dot product is zero exactly when vectors are perpendicular, this formula also doubles as the standard test for orthogonality.