The angle between two vectors in 2D or 3D — or the acute angle between two lines from their slopes.
angle (degrees)
How to use this tool
Two vectors in 2D or 3D, or two lines given by their slopes.
Vector components, or the two slopes m1 and m2.
Both are shown, so the answer works for homework and for code.
Perpendicular vectors give exactly 90°, parallel vectors 0° (or 180° if opposite).
Formula & worked example
(1, 0) and (0, 1) are perpendicular: 90°. In 3D, (1, 2, 3) and (4, 5, 6) give cos θ = 32 ÷ √(14 × 77) ≈ 0.9746, so θ ≈ 12.93°.
Limitations — read this before you rely on it
The vector angle is always the smaller angle between the vectors (0° to 180°). For lines, the acute angle (0° to 90°) is returned; perpendicular lines are exactly 90°.
Frequently asked questions
Use the dot product: cos θ = (a·b) ÷ (|a| × |b|), then take arccos. This calculator does all three steps.
90° — they are perpendicular, and their dot product is 0.
Yes — switch to 3D mode and enter z components. The same dot-product formula extends naturally.
From slopes: tan θ = |(m2 − m1) ÷ (1 + m1m2)|. Slopes 1 and −1 give 90° (perpendicular lines).
Because the angle between vectors is defined as the smaller rotation that takes one to the other, from 0° to 180°.