If A and B are the diagonals of a parallelogram, find a vector formula for the area of the parallelogram.

It has been given that A and B are the diagonal of the parallelogram.

A quantity that has magnitude as well as direction is called a vector. It is typically denoted by an arrow in which the head determines the direction of the vector and the length determines it magnitude.

The area of a parallelogram that has diagonals of lengths p and $q$with angle$\theta$between them is given by the formula mentioned below.

$\frac{1}{2}pq\sin \left(\theta \right)$

This is the same as the formula for cross product except for an extra $\frac{1}{2}$in front. Thus, write the vector form of the area of a parallelogram with diagonals A andB.

$\frac{1}{2}\left|A\times B\right|$

It must be known that$\frac{1}{2}\left|A\times B\right|=\frac{1}{2}\left|A\times B\right|$since the only difference is a minus sign and this does not matter in the magnitude.

Hence, the vector formula for the area of the parallelogram is$\frac{1}{2}\left|A\times B\right|$.