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
\(\frac{\vec{PQ}}{|\vec{PQ}|}=\frac{3\hat{i}+3\hat{j}+3\hat{k}}{3\sqrt{3}}=\frac{1}{\sqrt3\hat{i}}+\frac{1}{\sqrt{3}\hat{j}}+\frac{1}{\sqrt{3}\hat{k}}.\)