Overview
Two vectors in space have two products: the dot product, which yields a scalar, and the cross product, which yields a vector perpendicular to both. The dot product measures alignment, while the cross product measures the area of the parallelogram they form.
Formula
A · B = |A||B| cos θ, and |A × B| = |A||B| sin θ.
How it works
Enter both vectors
Three components for each vector.
Compute the dot product
Multiply corresponding components and add.
Compute the cross product
Use the 3×3 determinant formula.
Compute the angle
Use cos θ = (A · B) ÷ (|A||B|).
Worked examples
Example 1: A = (1, 2, 3) and B = (4, 5, 6)
A · B = 32, and A × B = (−3, 6, −3).
Example 2: A = (1, 0, 0) and B = (0, 1, 0)
A · B = 0, and A × B = (0, 0, 1).
Frequently asked questions
What does a zero dot product mean?
The vectors are perpendicular, since cos 90° = 0.
Is the cross product commutative?
No, A × B = −(B × A).
How do I find the cross product direction?
Use the right-hand rule; the result is perpendicular to the plane containing both vectors.
This calculator is for educational and informational purposes only and is not a substitute for professional advice.