Angle Between Vectors

The angle between vectors calculator finds the cosine of the angle and the angle itself between two vectors from their coordinates in the plane and in space, as well as angle ABC of a triangle from the coordinates of three points. The cosine is written exactly (for example, −2√6/21), standard angles are exact (60°, 135°), others are in degrees to 0.0001.

Formula for the angle between vectors

cos φ = (a · b) / (|a| · |b|) = (x₁x₂ + y₁y₂ + z₁z₂) / (√(x₁² + y₁² + z₁²) · √(x₂² + y₂² + z₂²))

The angle between vectors is always from 0° to 180°.

Example 1. a = (1, 1, 0), b = (0, 1, 1).
a · b = 1, |a| = |b| = √2, cos φ = 1/2 ⇒ φ = 60°.

Example 2. Angle ABC: A(1, 0), B(0, 0), C(1, 1).
BA = (1, 0), BC = (1, 1), cos ∠B = 1/√2 = √2/2 ⇒ ∠ABC = 45°.

The type of angle without calculation

If the vectors point the same way, the angle is 0°; if they point in opposite directions, it is 180°. For a zero vector the angle is undefined.