If the vectors → a = λ ^ i + μ ^ j + 4 ^ k , → b = - 2 ^ i + 4 ^ j...

Given,

\overset{\to}{a} = \lambda\overset{^}{i} + \mu\overset{^}{j} + 4\overset{^}{k}, \overset{\to}{b} = - 2\overset{^}{i} + 4\overset{^}{j} - 2\overset{^}{k} and \overset{\to}{c} = 2\overset{^}{i} + 3\overset{^}{j} + \overset{^}{k} are coplanar

So, \left|$\begin{cases}\lambda & \mu & 4 \\ - 2 & 4 & - 2 \\ 2 & 3 & 1\end{cases}$\right| = 0$</p><p>$⇒ \lambda ( 10 ) - \mu ( 2 ) + 4 ( - 14 ) = 0$</p><p>$⇒ 10\lambda - 2\mu = 56$</p><p>$⇒ 5\lambda - \mu = 28 . . . . . \left(1\right)$</p><p>And given projection of$\overset{\to}{a}$on$\overset{\to}{b}$is$\sqrt{54}$</p><p>$⇒ \dfrac{\overset{\to}{a} \cdot \overset{\to}{b}}{| \overset{\to}{b} |} = \sqrt{54}$</p><p>$⇒ \dfrac{- 2\lambda + 4\mu - 8}{\sqrt{24}} = \sqrt{54}$</p><p>$⇒ - 2\lambda + 4\mu - 8 = \sqrt{54 \times 24} . . . . \left(2\right)$</p><p>By solving equation \left(1\right) \text{ and } \left(2\right)$ we get,

$⇒ \lambda + \mu = 24$