Dot product calculator
Calculator
The calculator needs JavaScript. The tables below give the same numbers for common cases.
How it is worked out
- Multiply matching components and add: A · B = AxBx + AyBy + AzBz = 12.
- The angle follows from cos θ = (A · B) ÷ (|A| |B|) = 0.365487, so θ = 68.562447°.
| Inputs | Answer |
|---|---|
| Vector A: x: 1; y: 2; z: 3; Vector B: x: 4; y: -5; z: 6 | A · B = (1)(4) + (2)(-5) + (3)(6) = 12. |
| Vector A: x: 1; y: 0; z: 0; Vector B: x: 0; y: 1; z: 0 | A · B = (1)(0) + (0)(1) + (0)(0) = 0. |
| Vector A: x: 2; y: 3; z: 0; Vector B: x: 4; y: 1; z: 0 | A · B = (2)(4) + (3)(1) + (0)(0) = 11. |
How to use it
The dot product of two vectors multiplies matching components and adds them: A · B = AxBx + AyBy + AzBz. It is also |A||B|cos θ, where θ is the angle between the vectors, so the dot product is 0 exactly when the vectors are perpendicular.
A positive dot product means the angle is under 90°, and a negative one means it is over 90°. The angle follows from cos θ = (A · B) ÷ (|A||B|), which needs both vectors to be non-zero.
Frequently asked questions
What is the dot product of two vectors?
The sum of the products of matching components.
When is the dot product zero?
When the vectors are perpendicular (or one is the zero vector).
How do I find the angle between two vectors?
cos θ = (A · B) ÷ (|A||B|), then take the inverse cosine.
Method: standard arithmetic shown step by step; no outside data is used. Checked 2026-11-12.
About · How we verify this · Editorial policy · Corrections
General information only. Results are plain arithmetic on the numbers you enter; check anything important (money, health, school work) against your own records or the official source.