Question

$P$ and $Q$ are two non-zero vectors inclined to each other at an angle $'\theta'$. $'p'$ and $'q'$ 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 }{P Q} \Rightarrow \frac{Q_{P}}{Q}=\frac{ P \cdot Q }{P Q} $
$\Rightarrow Q_{P}=\frac{P \cdot Q}{P} \,\,\,\,\,\,\,...(ii)$
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
$Q_{p}=\hat{P} \cdot Q$