Question

What is a Dot Product Geometrically?

Answer 1

Geometrical interpretation of Dot product

A dot product is the product of the magnitude of two vectors and the cosecant of the angle between the vectors.

$\overrightarrow{a}.\overrightarrow{b}=\left|\overrightarrow{a}\right|\left|\overrightarrow{b}\right|\cos \theta$

Consider the vectors $\overrightarrow{a}$ and $\overrightarrow{b}$and the angle between the vectors is $\theta$. Draw $BL\perp OA$and $AM\perp OB$

In $△OBL,$

$\cos \theta =\frac{OL}{OB}$

$\Rightarrow \cos \theta =\frac{OL}{\left|\overrightarrow{a}\right|}$

$\Rightarrow OL=\overrightarrow{\left|a\right|}\cos \theta .............\left(1\right)$

The dot product is

$\overrightarrow{a}·\overrightarrow{b}=\left|\overrightarrow{a}\right|\left|\overrightarrow{b}\right|\cos \theta$

$\Rightarrow \overrightarrow{a}·\overrightarrow{b}=\left|\overrightarrow{b}\right|OL\left[\mathrm{substituting}\mathrm{from}\mathrm{equation}\left(1\right)\right]$

Similarly $\overrightarrow{a}·\overrightarrow{b}=\left|\overrightarrow{a}\right|OM$

Hence,

The dot product is the dot product of the magnitude of the vector $\overrightarrow{b}$and the projection of $\overrightarrow{a}$ on $\overrightarrow{b}$or the dot product is the dot product of the magnitude of the vector $\overrightarrow{a}$and the projection of $\overrightarrow{b}$ on $\overrightarrow{a}$.