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# Find a unit vector in the direction of $\overrightarrow{PQ},$where $P$ and $Q$ have co-ordinates $(5,0,8)$ and $(3,3,2)$,respectively.

$\begin{array}{1 1} (A)\;\Large\frac{2\hat i+3\hat j+6\hat k}{7} \\ (B)\;\Large\frac{-2\hat i+3\hat j-6\hat k}{7} \\(C)\;\Large\frac{-2\hat i+3\hat j-6\hat k}{49} \\(D)\;\Large\frac{-2\hat i-3\hat j+6\hat k}{7} \end{array}$

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A)
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• Unit vector in the direction of $\overrightarrow a=\large\frac{\overrightarrow a}{|\overrightarrow a|}$
• $\overrightarrow{PQ}=\overrightarrow{OQ}-\overrightarrow{OP}$
• $|x \hat i+y \hat j+2 \hat k|=\sqrt {x^2+y^2+z^2}$
Given $P(5,0,8)\; and\; Q(3,3,2)$
Let $\overrightarrow{OP}=5\hat i+8\hat k\:and\:\overrightarrow{OQ}=3\hat i+3\hat j+2\hat k$
We know that $\overrightarrow{PQ}=\overrightarrow{OQ}-\overrightarrow{OP}$
$\overrightarrow{PQ}=(3\hat i+3\hat j+2\hat k)-(5\hat i+8\hat k)$
$\qquad =-2\hat i+3\hat j-6\hat k$
Magnitude of $PQ$ is
$|\overrightarrow{PQ}|=\sqrt{(-2)^2+3^2+(-6)^2}$
$\quad\quad =\sqrt {4+9+36}$
$\quad\quad =\sqrt {49}=7$
Unit vector in the direction of $\hat{PQ}=\frac{\overrightarrow{PQ}}{|\overrightarrow{PQ}|}$
$\qquad =\Large\frac{-2\hat i+3\hat j-6\hat k}{7}$