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 = < , , >
v = < , , >


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 = bf - ce
y = cd - af
z = ae - bd

Example 1
Let u = < 2 , 2 , 0 > and v = < 2 , - 2 , 0 >
w = u v = < x , y , z >
x = 20 - 0(-2) = 0
y = 02 - 20 = 0
z = 2(-2) - 22 = -8
Hence w = < 0 , 0 , -8 >
Below are shown the plots of vectors u , v and w = u v

Orthogonal 3 Dimensional Vectors

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.
Cross Product and Area of Parallelogram


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.