# Cross Product of Two Vectors - Calculator

This online calculator calculates the cross product of two vectors given by their components.

## Use of Cross Product Calculator

1 - Enter the components of each of the two vectors, as real numbers in decimal from and press "Calculate Cross Product". The answer is a vector w.
No characters other than real numbers are accepted by the calculator.

 u = < 2 , 2 , 0 > v = < 2 , -2 , 0 > w = < , , , >

## Definition of Cross Product of two Vectors

Let u and v be two vectors given by their 3 components as follows
u = < a , b , c > and v = < d , e , f >
The cross product of the two vectors u and v, given above, is another vector w given by
w = u × v = < a , b , c > × < d , e , f > = < x , y , z >
with the components x, y and z of vector w given by:
x = b×f - c×e
y = c×d - a×f
z = a×e - b×d

Example 1
Let u = < 2 , 2 , 0 > and v = < 2 , - 2 , 0 >
w = u × v = < x , y , z >
x = 2×0 - 0×(-2) = 0
y = 0×2 - 2×0 = 0
z = 2×(-2) - 2×2 = -8
Hence w = < 0 , 0 , -8 >
Below are shown the plots of vectors u , v and w = u ×v

## Applications of the Cross Product

The cross product of two vectors has applications in mathematics, physics, engineering, ... Some examples are presented below.
Example 2
One of the formulas to find the area of a triangle with sides having lengths |u| and |v| and an angle between then equal to θ is
(1 / 2) |u| . |v| sin(θ)
Also, it can be shown that that
| u × v | = |u||v| sin(θ)
We can therefore conclude that the area of a parallelogram spanned by vectors U and v is equal to the magnitude of the cross product of vectors u and v.

Example 3 - Application of Cross Product in Physics
When a particle of charge q is moving at a velocity v in a magnetic field B, a force F acts on this charge and is given by
F = q (v × B)
q is a scalar , v and B are vectors, v × B is the cross product of v and B.