Vector Dot Product Calculator
Calculate the dot product of two vectors and the angle between them.
- Free to use
- Accurate results
- No registration required
- Works on all devices
Enter two vectors to find their dot product and the angle between them.
Components separated by commas — works for 2D, 3D, or higher dimensions.
Your result
32
Dot product
Angle between vectors12.933
AI explanation
Formula
a·b = sum of (a_i × b_i); angle = arccos((a·b) ÷ (|a| × |b|))Worked example
(1,2,3) . (4,5,6)
| Field | Value |
|---|---|
| Vector A | 1,2,3 |
| Vector B | 4,5,6 |
| Dot product | 32 |
| Angle between vectors | 12.9333 |
Assumptions
- Both vectors must have the same number of components.
- The angle is undefined for a zero vector (magnitude 0).
Frequently asked questions
What does the dot product tell you?
It's a single number that combines the vectors' magnitudes and the angle between them — zero means the vectors are perpendicular, and it's positive when they point in a broadly similar direction.
How is the angle between two vectors found?
From the dot product formula rearranged: cos(θ) = (a·b) ÷ (|a| × |b|), then θ = arccos of that value.
Does the dot product work in any dimension?
Yes — unlike the cross product (which is specific to 3D), the dot product works for vectors of any equal dimension.