The dot product measures how much two vectors point in the same direction. Zero means they are perpendicular.
How to use the Dot Product and Angle Between Vectors Calculator
- a₁
- a₂
- a₃
- b₁
- b₂
- b₃
Results update as you type, and every field accepts the units shown in its menu.
How it works
Multiply matching components and add them. Divide by both magnitudes and take the arccosine for the angle.
Inputs explained
| Field | What to enter |
|---|---|
| a₁ | A number |
| a₂ | A number |
| a₃ | A number |
| b₁ | A number |
| b₂ | A number |
| b₃ | A number |
What each result means
- Dot product = [a₁]×[b₁] + [a₂]×[b₂] + [a₃]×[b₃]
- |a| = sqrt([a₁]^2 + [a₂]^2 + [a₃]^2)
- |b| = sqrt([b₁]^2 + [b₂]^2 + [b₃]^2)
- Angle between vectors = deg(acos([Dot product]÷([|a|]×[|b|])))
- Cross product x = [a₂]×[b₃] – [a₃]×[b₂]
- Cross product y = [a₃]×[b₁] – [a₁]×[b₃]
- Cross product z = [a₁]×[b₂] – [a₂]×[b₁]
Example values
How the main result changes when a₁ changes and everything else stays at the example values above:
| a₁ | Dot product |
|---|---|
| 0.5 | 10 |
| 0.75 | 11 |
| 1 | 12 |
| 1.25 | 13 |
| 1.5 | 14 |
| 2 | 16 |
Formula
a · b = a₁b₁ + a₂b₂ + a₃b₃θ = arccos(a · b ÷ (|a||b|))Worked example
Inputs — a₁: 1; a₂: 2; a₃: 3; b₁: 4; b₂: -5; b₃: 6.
Results — Dot product = 12; |a| = 3.741657; |b| = 8.774964; Angle between vectors = 68.5624 °; Cross product x = 27; Cross product y = 6; Cross product z = -13.
Frequently asked questions
What does a negative dot product mean?
The angle between the vectors is more than 90°.
What is the cross product?
A vector perpendicular to both, also shown here.
What formula does the Dot Product and Angle Between Vectors Calculator use?
a · b = a₁b₁ + a₂b₂ + a₃b₃.