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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 $\;2\overrightarrow{ a}- \overrightarrow{b}$

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• 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 $2\overrightarrow{a}-\overrightarrow{b}=2(\hat i+\hat j+2 \hat k)-(2 \hat i+\hat j-2 \hat k)$
$\qquad\qquad \qquad= 2\hat i+2\hat j+4 \hat k-2 \hat i-\hat j+2 \hat k$
Therefore $2\overrightarrow{a}-\overrightarrow{b}=\hat j+6 \hat k$
The magnitude of this vector is
$|2\overrightarrow{a}-\overrightarrow{b}|=\sqrt {(1)^2+6^2}$
$=\sqrt {37}$
Hence the Unit vector in the direction of $|(2 \overrightarrow {a}-\overrightarrow {b})| is =\large \frac{2\overrightarrow {a}-\overrightarrow {b}}{|2\overrightarrow {a}-\overrightarrow {b}|}$
$=\Large\frac{\hat j+6 \hat k}{\sqrt {37}}$