$$\overrightarrow a = \alpha \widehat i + 3\widehat j - \widehat k$$

$$\overrightarrow b = 3\widehat i - \beta \widehat j + 4\widehat k$$

$$\overrightarrow c = \widehat i + 2\widehat j - 2\widehat k$$

Projection of $$\overrightarrow a $$ on $$\overrightarrow c $$ is

$${{\overrightarrow a \,.\,\overrightarrow c } \over {|\overrightarrow b |}} = {{10} \over 3}$$

$${{\alpha + 6 + 2} \over {\sqrt {{1^2} + {2^2} + {{( - 2)}^2}} }} = {{\alpha + 8} \over 3} = {{10} \over 3}$$

$$\therefore$$ $$\alpha$$ = 2

$$\overrightarrow b \times \overrightarrow c = - 6\widehat i + 10\widehat j + 7\widehat k$$

$$\left| {\matrix{ {\widehat i} & {\widehat j} & {\widehat k} \cr 3 & { - \beta } & 4 \cr 1 & 2 & { - 2} \cr } } \right| = (2\beta - 8)\widehat i + 10\widehat j + (6 + \beta )\widehat k = - 6\widehat i + 10\widehat j + 7\widehat k$$

$$2\beta - 8 = - 6$$ & $$6 + \beta = 7$$

$$\therefore$$ $$\beta$$ = 1

$$\alpha + \beta = 2 + 1 = 3$$