Let $\overrightarrow a = \alpha \widehat i + 3\widehat j - \widehat k$, $\overrightarrow b = 3\widehat i - \beta \widehat j + 4\widehat k$ and $\overrightarrow c = \widehat i + 2\widehat j - 2\widehat k$ where $\alpha ,\,\beta \in R$, be three vectors. If the projection of $\overrightarrow a$ on $\overrightarrow c$ is ${{10} \over 3}$ and $$\overrightarrow b \times \overrightarrow c = - 6\widehat i + 10\widehat j + 7\widehat k$$, then the value of $\alpha + \beta$ is equal to :

<p>$\overrightarrow a = \alpha \widehat i + 3\widehat j - \widehat k$</p> <p>$\overrightarrow b = 3\widehat i - \beta \widehat j + 4\widehat k$</p> <p>$\overrightarrow c = \widehat i + 2\widehat j - 2\widehat k$</p> <p>Projection of $\overrightarrow a$ on $\overrightarrow c$ is</p> <p>$${{\overrightarrow a \,.\,\overrightarrow c } \over {|\overrightarrow b |}} = {{10} \over 3}$$</p> <p>$${{\alpha + 6 + 2} \over {\sqrt {{1^2} + {2^2} + {{( - 2)}^2}} }} = {{\alpha + 8} \over 3} = {{10} \over 3}$$</p> <p>$\therefore$ $\alpha$ = 2</p> <p>$$\overrightarrow b \times \overrightarrow c = - 6\widehat i + 10\widehat j + 7\widehat k$$</p> <p>$$\left| {\matrix{ {\widehat i} & {\widehat j} & {\widehat k} \cr 3 & { - \beta } & 4 \cr 1 & 2 & { - 2} \cr } } \right| = (2\beta - 8)\widehat i + 10\widehat j + (6 + \beta )\widehat k = - 6\widehat i + 10\widehat j + 7\widehat k$$</p> <p>$2\beta - 8 = - 6$ & $6 + \beta = 7$</p> <p>$\therefore$ $\beta$ = 1</p> <p>$\alpha + \beta = 2 + 1 = 3$</p>