Let two vectors P and Q are represented by graph as below

Here, $Q_p$ is a vector in the direction of P.
Then, from the right angle triangle, we get
$\mathrm{c}\mathrm{o}\mathrm{s}\theta =\frac{Q_p}{Q}$ ….(i)
$\Rightarrow Q_p=Q\mathrm{c}\mathrm{o}\mathrm{s}\theta$
Also, $\mathrm{c}\mathrm{o}\mathrm{s}\theta =\frac{P.Q}{PQ}\Rightarrow \frac{Q_p}{Q}=\frac{P.Q}{PQ}$
$\Rightarrow Q_p=\frac{P.Q}{P}$ ….(ii)
As given that, P is the unit vector along P, then
$\widehat{P}=\frac{P}{P}$ …(iii)
Putting the value of P from Eq.(iii) to Eq.(ii), we get
$Q_p=P.Q$