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)

Then,

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 )

If,

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

Then,

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)