$A vector of magnitude 5 \sqrt{5} coplanar with vectors \hat{i} + 2 \hat{j} and \hat{j} + 2 \hat{k} and the$ $perpendicular vector \hat{2} i + \hat{j} + 2 \hat{k} is :$

Moderate

A

$\overset{}{\pm 5 (5 \overset{\land}{i}} + 6 \overset{\land}{j} - 8 \overset{\land}{k})$
B

$\overset{}{\pm \sqrt{5} (5 \overset{\land}{i}} + 6 \overset{\land}{j} - 8 \overset{\land}{k})$
C

$\overset{}{\pm 5 \sqrt{5} (5 \overset{\land}{i}} + 6 \overset{\land}{j} - 8 \overset{\land}{k})$
D

$\overset{}{\pm (5 \overset{\land}{i}} + 6 \overset{\land}{j} - 8 \overset{\land}{k})$

Solution

$Unit vector coplanar with \vec{a} and \vec{b} and perpendicular to$
$v e c t o r \vec{c} i s \pm \frac{(\vec{a} \times \vec{b}) \times \vec{c}}{| (\vec{a} \times \vec{b}) \times \vec{c} |}$
$\vec{a} \times \vec{b} = |\begin{matrix} i & j & k \\ 1 & 2 & 0 \\ 0 & 1 & 2 \end{matrix}| = \hat{i} (4 - 0) - \hat{j} (2 - 0) + \hat{k} (1 - 0) = 4 \hat{i} - 2 \hat{j} + \hat{k}$
$(\vec{a} \times \vec{b}) \times \vec{c} = |\begin{matrix} i & j & k \\ 4 & - 2 & 1 \\ 2 & 1 & 2 \end{matrix}| = \hat{i} (- 4 - 1) - \hat{j} (8 - 2) + \hat{k} (4 + 4)$
$= - 5 \hat{i} - 6 \hat{j} + 8 \hat{k} |(\vec{a} \times \vec{b}) \times \vec{c}| = \sqrt{125} = 5 \sqrt{5}$
$∴ u n i t v e c t o r o f m a g n i t u d e 5 \sqrt{5} c o p l a n a r i s \frac{5 \sqrt{5} [\pm (\vec{a} \times \vec{b}) \times \vec{c}]}{5 \sqrt{5}} = \pm (5 \hat{i} + 6 \hat{j} - 8 \hat{k})$

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