Question

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

• A. $\frac{1}{\sqrt{3}}\hat{i}+\frac{1}{\sqrt{3}}\hat{j}+\frac{2}{\sqrt{3}}\hat{k}$
• B. $\frac{1}{\sqrt{3}}\hat{j}+\frac{1}{\sqrt{3}}\hat{j}+\frac{1}{\sqrt{3}}\hat{k}$
• C. $\frac{1}{\sqrt{3}}\hat{i}+\frac{1}{\sqrt{5}}\hat{j}+\frac{1}{\sqrt{7}}\hat{k}$
• D. None of these

Answer 1

Answer: (B)

A vector with initial point $\left(x_1,y_1,z_1\right)$ and terminal point $\left(x_2,y_2,z_2\right)$ is given by $\left(x_2-x_1\right)\hat{i}+\left(y_2-y_1\right)\hat{j}+\left(z_2-z_1\right)\hat{k}$
The given points are $P(1,2,3)$ and $Q(4,5,6)$.
$\therefore x_1=1,y_1=2,z_1=3$ and $x_2=4,y_2=5,z_2=6$
So, vector $PQ=\left(x_2-x_1\right)\hat{i}+\left(y_2-y_1\right)\hat{j}+\left(z_2-z_1\right)\hat{k}$
$=(4-1)\hat{i}+(5-2)\hat{j}+(6-3)\hat{k}=3\hat{i}+3\hat{j}+3\hat{k}$
$\therefore$ 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 $PQ$,
$\frac{PQ}{|PQ|}=\frac{3\hat{i}+3\hat{j}+3\hat{k}}{3\sqrt{3}}=\frac{3}{3\sqrt{3}}(\hat{i}+\hat{j}+\hat{k})$
$=\frac{1}{\sqrt{3}}\hat{i}+\frac{1}{\sqrt{3}}\hat{j}+\frac{1}{\sqrt{3}}\hat{k}$