Question

Why does cross product give area of a parallelogram ?

Answer 1

Cross product

1. If $\overrightarrow{a}$ and $\overrightarrow{b}$are two vectors, their cross product is represented by $\overrightarrow{a}\times \overrightarrow{b}$ .
2. $\overrightarrow{a}\times \overrightarrow{b}=\left|a\right|\left|b\right|\sin \theta$, where $\theta$ is the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$.
3. It is the product of two vectors in a three-dimensional space.
4. The resultant is a vector having both magnitude and direction.

Area of a parallelogram

1. Area of a parallelogram is given by,
2. $Area=base\times altitude$
3. We know, $y$ component of $\overrightarrow{a}$ is $\left|a\right|\sin \theta$ and $x$ component is $\left|a\right|\cos \theta$.
4. Therefore, the altitude is $\left|a\right|\sin \theta$ and the base is$\left|b\right|$.
5. where $\theta$ is the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$.
6. Therefore, $area=\left|a\right|\left|b\right|\sin \theta$.

This is the reason cross product gives the area of parallelogram