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Angle Between Two Vectors Calculator

This angle between two vectors calculator is a useful tool for finding the angle between two 2D or 3D vectors.

Comma separated, 2D or 3D.
Clear
Angle45°0.78539816 radians
Dot product1
|a|1
|b|1.414214
cos θ0.70710678
Relationshipoblique
Cross product(0, 0, 1)magnitude 1

The rule

cos θ = (a · b) ÷ (|a| |b|)

Degrees and radians

A full turn is 360° or 2π radians, so:

radians = degrees × π/180 and degrees = radians × 180/π

Degrees are convenient because 360 divides so many ways. Radians are the natural choice in mathematics because arc length becomes simply rθ and the calculus of sine and cosine comes out clean — derivatives pick up stray factors of π/180 in degrees.

Names worth knowing

  • Acute — under 90°. Right — exactly 90°. Obtuse — between 90° and 180°.
  • Straight — 180°. Reflex — over 180°.
  • Complementary pairs add to 90°; supplementary pairs add to 180°.