Vector Dot Product Calculator
Turning Two Vectors Into a Single Number
Unlike vector addition, the dot product doesn't produce another vector — it collapses two vectors down into a single scalar that captures how much they point in the same direction. A positive result means the vectors broadly align, a negative result means they point broadly opposite, and zero means they're perpendicular.
The Formula
2D: A · B = (Ax×Bx) + (Ay×By)
3D: A · B = (Ax×Bx) + (Ay×By) + (Az×Bz)
3D: A · B = (Ax×Bx) + (Ay×By) + (Az×Bz)
Worked Example
For vectors A = (3, 4) and B = (1, 2):
| Step | Calculation | Result |
|---|---|---|
| Ax×Bx | 3 × 1 | 3 |
| Ay×By | 4 × 2 | 8 |
| Sum | 3 + 8 | 11 |
Where This Matters
- Work in physics — mechanical work equals the dot product of a force vector and a displacement vector.
- Lighting in 3D graphics — the brightness of a surface depends on the dot product between its normal vector and the direction to the light source.
- Testing perpendicularity — a dot product of exactly zero confirms two vectors are at right angles, a quick check used throughout geometry and engineering.
How to Use This Calculator
- Choose the dimension: 2D Vectors or 3D Vectors.
- Enter Vector A's x, y (and z for 3D) components.
- Enter Vector B's x, y (and z for 3D) components.
- Select Calculate to get the dot product.
Related Calculations
Use the dot product result to find an angle directly with the Angle Between Vectors Calculator, or compute a vector result instead with the Vector Cross Product Calculator.