Question

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

Answer 1

The given points are P(1,2,3) and Q(4,5,6).
$\therefore x_1=1,y_1=2,z_1=3and{x}_2=4,y_2=5,z_2=6$
So vector $PQ=(x_2-x_1)\hat{i}+(y_2-y_1)\hat{j}+(z_2-z_1)\hat{k}=(4-1)\hat{i}+(5-2)\hat{j}+(6-3)\hat{k}=3\hat{i}+3\hat{j}+3\hat{k}$
Comparing with $X=x\hat{i}+y\hat{j}+z\hat{k},wegetx=3,y=3,z=3$
$\therefore$ Magnitude of given vector
$|PQ|=\sqrt{3^2+3^2+3^2}=\sqrt{9+9+9}=\sqrt{27}=3\sqrt{3}$
Hence, the unit vector in the direction of PQ,
$\frac{PQ}{|PQ|}=\frac{3\hat{i}+3\hat{j}+3\hat{k}}{3\sqrt{3}}=\frac{3}{3\sqrt{3}}(\hat{i}+\hat{j}+\hat{k})=\frac{1}{\sqrt{3}}\hat{i}+\frac{1}{\sqrt{3}}\hat{j}+\frac{1}{\sqrt{3}}\hat{k}.$