Question

Prove that the area of the parallelogram is equal to | A × B |

Answer 1

The area of a paralellogram with base a and height h is given by:

`A=h\cdot a`

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:

`A=a\cdot b\cdot\sin (\phi)`

On the other hand, the cross product of two vectors is defined as:

`\vec{a}*\vec{b}=a\cdot b\cdot\sin (\phi)\hat{n}`

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:

`|\vec{a}*\vec{b}|=a\cdot b\cdot\sin (\phi)`

We can see that both the area of the parallelogram and the modulus of the cross product have the same expressions. Therefore:

`A=|\vec{a}*\vec{b}|`