Let → a = ^ i + ^ j + 2 ^ k , → b = 2 ^ i - 3 ^ j + ^ k and → c = ^...

Since vector $\overset{\to}{v}, \overset{\to}{a} \text{\text{ and } } \overset{\to}{b}$ are coplanar then, $\overset{\to}{v} = {\lambda}_{1}\overset{\to}{a} + {\lambda}_{2}\overset{\to}{b}$, where ${\lambda}_{1}, {\lambda}_{2} \in \mathbb{R}$.

⇒ \overset{\to}{v} = \left({\lambda}_{1} + 2{\lambda}_{2}\right)\overset{^}{i} + \left({\lambda}_{1} - 3{\lambda}_{2}\right)\overset{^}{j} + \left(2{\lambda}_{1} + {\lambda}_{2}\right)\overset{^}{k}

$∵$ Projection of $\overset{\to}{v}$ on $\overset{\to}{c}$ is $\dfrac{2}{\sqrt{3}}$

$∴ \dfrac{{\lambda}_{1} + 2{\lambda}_{2} - {\lambda}_{1} + 3{\lambda}_{2} + 2{\lambda}_{1} + {\lambda}_{2}}{\sqrt{3}} = \dfrac{2}{\sqrt{3}}$

$∴ {\lambda}_{1} + 3{\lambda}_{2} = 1 . . . \left(1\right)$

and \overset{\to}{v} \cdot \overset{^}{j} = 7 ⇒ {\lambda}_{1} - 3{\lambda}_{2} = 7 . . . \left(2\right)</p><p>⇒ \overset{\to}{v} = \left({\lambda}_{1} + 2{\lambda}_{2}\right)\overset{^}{i} + \left({\lambda}_{1} - 3{\lambda}_{2}\right)\overset{^}{j} + $\left(2{\lambda}_{1} + {\lambda}_{2}\right)$\overset{^}{k}$</p><p>$∴ {\lambda}_{1} + 3{\lambda}_{2} = 1 . . . \left(1\right)$</p><p>$⇒ {\lambda}_{1} - 3{\lambda}_{2} = 7\left(2\right)
From equation $\left(1\right)$ and \left(2\right), we get

${\lambda}_{1} = 4, {\lambda}_{2} = - 1$

∴ \overset{\to}{v} = 2\overset{^}{i} + 7\overset{^}{j} + 7\overset{^}{k}

∴ \overset{\to}{v} \cdot \left(\overset{^}{i} + \overset{^}{k}\right) = 2 + 7 = 9$</p>$