How to use

This vector calculator works with two 3D vectors A and B. It reports each magnitude, the dot product A·B, the cross product A×B and its magnitude, and the angle between the vectors when both are nonzero. A 2D vector can be represented by setting z=0.

Enter the x, y and z components of A.
Enter the components of B.
Calculate the magnitudes, dot product, cross product and angle.

How it is calculated

A·B=ΣAᵢBᵢ; |A|=√ΣAᵢ²; θ=acos((A·B)/(|A||B|))

Example

Example: A=(1,2,3), B=(4,5,6). A·B=32, A×B=(-3,6,-3), |A×B|=√54≈7.3485 and the angle is about 12.93°.

Important notes

If either vector is zero, its direction is undefined and so is the angle between the vectors; the calculator does not invent a zero-degree angle. The magnitude of A×B equals the area of the parallelogram spanned by A and B.

Frequently asked questions

When are vectors perpendicular?

When A·B=0, provided neither vector is zero.

What does the cross product give?

A vector perpendicular to A and B following the right-hand rule.

Can I use 2D vectors?

Yes. Set both z components to zero.

Is this calculator free?

Yes. It is free to use and requires no account.

Vector Calculator