# Cross Product of Two Vectors - Calculator

An online calculator that calculates the cross product of two vectors, given by their components, is presented.

## 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

## 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.

## 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 ? is

(1 / 2) |u| . |v| sin(?)

Also, it can be shown that that

| u **×** v | = |u||v| sin(?)

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.

## More References and Links

vectors.

Cross Product of 3D Vectors.