Cross products calculator of two vectors, cross product calculator. Calculate the cross product of two vectors for a given point A x B matrix. Learn properties of Cross Products.



How to use Cross Product Calculator?

The above Cross Product calculator can be used to find the cross product of two vectors easily.

Just fill in the values of vector A in the 3 input fields
Fill the values of vector B in the input fields.
Once you have entered the values, just click on the calculate button to get the coss product and see the result.

How to find the Cross Product of two Vectors?

The Cross product of vector A and B is defined as a vector such that

  • the magnitude is ab \sin \theta , Where \theta is the angle between A and B
  • Its direction is perpendicular to the plane of A and B
  • A right-handed system is formed with A and B

Let N be the unit vector normal to the plane of A and B (Where N, A, B are forming a right-handed system)


A \times B =ab \sin \theta N

Properties of Cross Products

-Vector product of two vectors is not commutative

A \times B  \neq  B \times A 

As a matter of fact, A \times B  =  -B \times A


-When the two vectors A and B are parallel to each other and the angle \theta is 0 or 180° So that Sin \theta = 0, and A \times B

An analytical expression for the vector product. (I,J,K be the Orthonormal Vectors )


A=a_{1}I+a_{2}J+a_{3}K & B=b_{1}I+b_{2}J+b_{3}K


A \times B= \begin{pmatrix}I& J&K\\ a_{1}& a_{2}&a_{3} \\b_{1}& b_{2}&b_{3} \end{pmatrix}

So, we get

A \times B = (a_{2}b_{3}-a_{3}b_{2})I+(a_{3}b_{1}-a_{1} b_{3})J+(a_{1}b_{2}-a_{2}b_{1})K

-Vector product of two vectors is distributive

A \times B \times C  = A \times C + B \times C

Example1: The Cross product of A=(1,2,3) and B=(4,5,6)

Solution Example:

A \times B = (a_{2}b_{3}-a_{3}b_{2})I+(a_{3}b_{1}-a_{1} b_{3})J+(a_{1}b_{2}-a_{2}b_{1})K 

I=a_{2}b_{3}-a_{3}b_{2}=2 \times 6-3 \times 5=12-15=-3 

J=a_{3}b_{1}-a_{1} b_{3}=3 \times 4-1 \times 6=12-6=6 

K= a_{1}b_{2}-a_{2}b_{1}=1 \times 5-2 \times 4=5-8=-3 

So, the cross product of A \times B= (-3,6,-3)