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# If $\overrightarrow{a}=\hat i-\hat j+2\hat k\;and\;\overrightarrow{b}=2\hat i-\hat j-2\hat k,$then find the unit vector in the direction of $6 \overrightarrow{b}$

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

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A)
• Unit vector in the direction of $\overrightarrow a=\large\frac{\overrightarrow a}{|\overrightarrow a|}$
Let $\overrightarrow a=\hat i-\hat j+2\hat k\:and\:\overrightarrow b=2\hat i-\hat j-2\hat k$
Therefore $6\overrightarrow b=12 \hat {i}-6 \hat {j}-12 \hat {k}$
$|6\overrightarrow b|=\sqrt {(12)^2+(6)^2+(-12)^2}$
$\quad\quad\quad=\sqrt{144+36+144}$
$\quad\quad\quad=\sqrt{324}=18$
Hence the Unit vector in the dirction of $6\overrightarrow b=\large\frac{6\overrightarrow b}{|6 \overrightarrow b|}$
$=\large\frac{12 \hat {i}-6 \hat {j}-12 \hat {k}}{18}$
$=\large\frac{1}{3}$$(2 \hat {i}- \hat {j}-2 \hat {k})$