Question

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

Answer 1

$\because P=(1,2,3)andQ=(4,5,6)\therefore \overrightarrow{PQ}=(4-1)\stackrel{\wedge }{i}+(5-2)\stackrel{\wedge }{j}+(6-3)\stackrel{\wedge }{k}=3\stackrel{\wedge }{i}+3\stackrel{\wedge }{j}+3\stackrel{\wedge }{k}$

vector joining $P$ and $Q$ is given by

$\overrightarrow{PQ}=3\stackrel{\wedge }{i}+3\stackrel{\wedge }{j}+3\stackrel{\wedge }{k}$

Magnitude of vector$\overrightarrow{PQ}$

$\left|\overrightarrow{PQ}\right|=\sqrt{3^2+3^2+3^2}=3\sqrt{3}$

therefore unit vector in direction of $\overrightarrow{PQ}$

$=\frac{\overrightarrow{PQ}}{\left|\overrightarrow{PQ}\right|}=\frac{3\stackrel{\wedge }{i}+3\stackrel{\wedge }{j}+3\stackrel{\wedge }{k}}{3\sqrt{3}}=\frac{1}{\sqrt{3}}\stackrel{\wedge }{i}+\frac{1}{\sqrt{3}}\stackrel{\wedge }{j}+\frac{1}{\sqrt{3}}\stackrel{\wedge }{k}$