Question

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

Short answer

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

Step-by-step solution

Given Information.

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

Definition of vector.

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.

Use the formula to evaluate the expression.

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