If the area of parallelogram, whose diagonals are $\hat{\mathrm{i}}-\hat{\mathrm{j}}+2 \hat{\mathrm{k}}$ and $2 \hat{\mathrm{i}}+3 \hat{\mathrm{j}}+\alpha \hat{\mathrm{k}}$ is $\frac{\sqrt{93}}{2}$ sq. units, then $\alpha=$

If the lengths of three vectors $\bar{a}, \bar{b}$ and $\bar{c}$ are $5,12,13$ units respectively, and each one is perpendicular to the sum of the other two, then $|\overline{\mathrm{a}}+\overline{\mathrm{b}}+\overline{\mathrm{c}}|=\ldots \ldots$.

The projection of the line segment joining $\mathrm{P}(2,-1,0)$ and $\mathrm{Q}(3,2,-1)$ on the line whose direction ratios are $1,2,2$ is

If $\bar{a}, \bar{b}, \bar{c}$ are three unit vectors such that $|\overline{\mathrm{a}}+\overline{\mathrm{b}}|^2+|\overline{\mathrm{a}}+\overline{\mathrm{c}}|^2=8$, then $|\overline{\mathrm{a}}+3 \overline{\mathrm{~b}}|^2+|\overline{\mathrm{a}}+3 \overline{\mathrm{c}}|^2=$