The area of a paralellogram with base a and height h is given by:
[tex]A=h\cdot a[/tex]If two adjacent sides of a parallelogram have lengths a and b and are separated by an angle φ, then the base of the parallelogram is a and the height is given by b*sin(φ). Then, the area of the parallelogram is given by:
[tex]A=a\cdot b\cdot\sin (\phi)[/tex]On the other hand, the cross product of two vectors is defined as:
[tex]\vec{a}\times\vec{b}=a\cdot b\cdot\sin (\phi)\hat{n}[/tex]Where the unitary vector is directed toward the direction perpendicular to a and b according to the right hand rule.
The modulus of the cross product of a and b is:
[tex]|\vec{a}\times\vec{b}|=a\cdot b\cdot\sin (\phi)[/tex]We can see that both the area of the parallelogram and the modulus of the cross product have the same expressions. Therefore:
[tex]A=|\vec{a}\times\vec{b}|[/tex]