Question

A unit vector in the direction of vector $\overrightarrow{PQ},$ where $P$ and $Q$ are the points $(1,2,3)$ and $(4,5,6)$ respectively is

• A. $\frac{1}{\sqrt{3}}\hat{i}+\frac{1}{\sqrt{3}}\hat{j}+\frac{1}{\sqrt{3}}\hat{k}$
• B. $\frac{1}{\sqrt{3}}\hat{i}+\frac{1}{\sqrt{3}}\hat{j}-\frac{1}{\sqrt{3}}\hat{k}$
• C. $2\hat{i}-\hat{j}+\hat{k}$
• D. None

Answer 1

The correct option is A $\frac{1}{\sqrt{3}}\hat{i}+\frac{1}{\sqrt{3}}\hat{j}+\frac{1}{\sqrt{3}}\hat{k}$

The points $P$ and $Q$ are $(1,2,3)$ and $(4,5,6)$ respectively.
$\overrightarrow{PQ}=(4-1)\hat{i}+(5-2)\hat{i}+(6-3)\hat{k}=3\hat{i}+3\hat{j}+3\hat{k}$,
$|\overrightarrow{PQ}|=\sqrt{3^2+3^2+3^2}=\sqrt{27}=3\sqrt{3}$
Therefore, unit vector in the direction of $\overrightarrow{PQ}$
$\widehat{PQ}=\frac{1}{|\overrightarrow{PQ}|}\overrightarrow{PQ}=\frac{1}{3\sqrt{3}}(3\hat{i}+3\hat{j}+\hat{k})$
$=\frac{1}{\sqrt{3}}\hat{i}+\frac{1}{\sqrt{3}}\hat{j}+\frac{1}{\sqrt{3}}\hat{k}$