Question

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

Answer 1

The correct answer is:$\frac{1}{\sqrt3\hat{i}}+\frac{1}{\sqrt{3}\hat{j}}+\frac{1}{\sqrt{3}\hat{k}}.$
The given points are $P(1,2,3)$,and $Q(4,5,6).$
$∴\vec{PQ}=(4-1)\hat{i}+(5-2)\hat{j}+(6-3)\hat{k}=3\hat{i}+3\hat{j}+3\hat{k}$
$|\vec{PQ}|=\sqrt{3^2+3^2+3^2}=\sqrt{9+9+9}=\sqrt{27}=3\sqrt{3}$
Hence,the unit vector in the direction of $\vec{PQ}$is
$\frac{\vec{PQ}}{|\vec{PQ}|}=\frac{3\hat{i}+3\hat{j}+3\hat{k}}{3\sqrt{3}}=\frac{1}{\sqrt3\hat{i}}+\frac{1}{\sqrt{3}\hat{j}}+\frac{1}{\sqrt{3}\hat{k}}.$