Let → a = α ^ i + 3 ^ j - ^ k , → b = 3 ^ i - β ^ j + 4 ^ k and → c...

Projection of $\overset{\to}{a}$ on $\overset{\to}{c}$ is $\dfrac{\overset{\to}{a} \cdot \overset{\to}{c}}{\left|\overset{\to}{c}\right|} = \dfrac{10}{3}$ (given)

$⇒ \dfrac{\alpha + 6 + 2}{\sqrt{1 + 4 + 4}} = \dfrac{10}{3} ⇒ \alpha = 2$

and \overset{\to}{b} \times \overset{\to}{c} = \left|\begin{cases}\overset{^}{i} & \overset{^}{j} & \overset{^}{k} \\ 3 & - \beta & 4 \\ 1 & 2 & - 2\end{cases}\right| = - 6\overset{^}{i} + \overset{^}{j} + \overset{^}{k}

$⇒ 2\beta - 8 = - 6 ⇒ \beta = 1$

Hence $\alpha + \beta = 3$