Find the unit vector in the direction of vector
Question:

Find the unit vector in the direction of vector $\overrightarrow{\mathrm{PQ}}$, where $\mathrm{P}$ and $\mathrm{Q}$ are the points

(1, 2, 3) and (4, 5, 6), respectively.

Solution:

The given points are P (1, 2, 3) and Q (4, 5, 6).

$\therefore \overrightarrow{\mathrm{PQ}}=(4-1) \hat{i}+(5-2) \hat{j}+(6-3) \hat{k}=3 \hat{i}+3 \hat{j}+3 \hat{k}$

$|\overrightarrow{\mathrm{PQ}}|=\sqrt{3^{2}+3^{2}+3^{2}}=\sqrt{9+9+9}=\sqrt{27}=3 \sqrt{3}$

Hence, the unit vector in the direction of $\overrightarrow{\mathrm{PQ}}$ is

$\frac{\overrightarrow{\mathrm{PQ}}}{|\overrightarrow{\mathrm{PQ}}|}=\frac{3 \hat{i}+3 \hat{j}+3 \hat{k}}{3 \sqrt{3}}=\frac{1}{\sqrt{3}} \hat{i}+\frac{1}{\sqrt{3}} \hat{j}+\frac{1}{\sqrt{3}} \hat{k} .$