The dot product multiplies matching components of two vectors and adds the results.
About this calculator
The dot product multiplies matching components of two vectors and adds the results. For (3,4)·(1,0) you get 3. The result is a single number — a scalar — not another vector.
The calculator also finds |a| and |b| and cos θ. When the dot product is zero, the vectors are perpendicular.
Dot products appear in work, projections, and lighting models. For matrix operations try the [[matrix]] calculator. For triangle side lengths use the [[pythagorean]] calculator. For slope between points see the [[slope]] calculator.
Common uses
- Check whether two vectors are perpendicular (dot = 0)
- Find cos θ for physics work W = F·d problems
- Compute projections in linear algebra homework
- Compare 2D and 3D vector alignment quickly
Frequently asked questions
What is (3,4)·(1,0)?
3. Only the x-components contribute: 3×1 + 4×0 = 3.
When is the dot product zero?
When vectors are orthogonal — perpendicular in 2D/3D. Example: (1,0)·(0,1) = 0.
How do I enter 3D vectors?
Switch to 3D and type three comma-separated numbers, e.g. 1, 2, 3 for each vector.
Is dot product the same as cross product?
No. Dot product returns a scalar; cross product (3D only) returns a vector perpendicular to both inputs.