Let \(\bar{a}=\alpha \hat{i}+3 \hat{j}-\hat{k}, \bar{b}=3 \hat{i}-\beta \hat{j}+4 \hat{k} \quad\) and \(\overline{\mathrm{c}}=\hat{\mathrm{i}}+2 \hat{\mathrm{j}}-2 \hat{\mathrm{k}}\), where \(\alpha, \beta \in \mathbb{R}\), be three vectors.
If the projection of \(\overline{\mathrm{a}}\) on \(\overline{\mathrm{c}}\) is \(\frac{10}{3}\) and \(\bar{b} \times \bar{c}=-6 \hat{i}+10 \hat{j}+7 \hat{k}\), then the value of \(2 \alpha+\beta\) is

Projection of \(\overline{\mathrm{a}}\) on \(\overline{\mathrm{c}}=\frac{\overline{\mathrm{a}} \cdot \overline{\mathrm{c}}}{|\overline{\mathrm{c}}|}\)
\(\begin{aligned}
& \therefore \quad \frac{\alpha+6+2}{\sqrt{1+4+4}}=\frac{10}{3} \\
& \therefore \quad \alpha=2 \\
& \overline{\mathbf{b}} \times \overline{\mathbf{c}}=\left|\begin{array}{ccc}
\hat{i} & \hat{j} & \hat{k} \\
3 & -\beta & 4 \\
1 & 2 & -2
\end{array}\right| \\
& =(2 \beta-8) \hat{i}-(-6-4) \hat{j}+(6+\beta) \hat{k} \\
& =(2 \beta-8) \hat{i}+10 \hat{j}+(6+\beta) \hat{k}
\end{aligned}\)
Comparing with \(-6 \hat{\mathrm{i}}+10 \hat{\mathrm{j}}+7 \hat{\mathrm{k}}\), we get
\(\begin{array}{ll}
& 2 \beta-8=-6 \\
\therefore & 2 \beta=2 \\
\therefore \quad & \beta=1 \\
\therefore \quad & 2 \alpha+\beta=5
\end{array}\)