Question

$P$ and $Q$ are two non-zero vectors inclined to each other at an angle ${}^{\prime }{\theta }^{\prime }$. ${}^{\prime }{p}^{\prime }$ and ${}^{\prime }{q}^{\prime }$ are unit vectors along $P$ and $Q$ respectively. The component of $Q$ in the direction of $Q$ will be

• A. $P\cdot Q$
• B. $\frac{P\times Q}{P}$
• C. $\frac{P\cdot Q}{Q}$
• D. $p\cdot q$

Answer 1

Answer: (A)

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
$\cos \theta =\frac{Q_p}{Q}...(i)$
$\Rightarrow Q_p=Q\cos \theta$
Also, $\cos \theta =\frac{P\cdot Q}{PQ}\Rightarrow \frac{Q_P}{Q}=\frac{P\cdot Q}{PQ}$
$\Rightarrow Q_P=\frac{P\cdot Q}{P}...(ii)$
As given that, $\hat{P}$ is the unit vector along $P$, then
$\hat{P}=\frac{P}{P}..(iii)$
Putting the value of Pfrom Eq. (iii) to Eq. (ii), we get
$Q_p=\hat{P}\cdot Q$