Dot Product
Find the dot product of vectors from coordinates or from lengths and the angle, the cosine of the angle, the projection and a perpendicularity check.
The dot product calculator finds the dot (scalar) product of vectors from their coordinates (in the plane and in space) or from their lengths and the angle between them. From coordinates it also finds the lengths, the cosine of the angle, the angle itself, the scalar projection of one vector onto the other, and checks perpendicularity.
Dot product formulas
| What is known | Formula |
|---|---|
| Coordinates in the plane | a · b = x1x2 + y1y2 |
| Coordinates in space | a · b = x1x2 + y1y2 + z1z2 |
| Lengths and angle | a · b = |a| · |b| · cos φ |
| Scalar projection of a onto b | compba = (a · b) / |b| |
Example 1. a = (1, 2, −3), b = (4, −1, 2).
a · b = 1 · 4 + 2 · (−1) + (−3) · 2 = −4 — the angle is obtuse.
Example 2. |a| = 3, |b| = 4, φ = 60°.
a · b = 3 · 4 · ½ = 6.
Properties
- a · b = 0 ⇔ the vectors are perpendicular (or one of them is zero).
- The sign shows the type of angle: positive — acute, negative — obtuse.
- a · a = |a|² — the square of a vector is the square of its length.
- The dot product is commutative and distributive: a · (b + c) = a · b + a · c.
In physics the dot product of force and displacement is the work done: W = F · s · cos φ.
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