Let a =α i +3 j - k, b =3 i -β j +4 k and c = i +2 j -2 k where α, β ∈ R, be three vectors. If the projection of a on c is 10/3 and b × c =-6 i +10 j…

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