The number of integral values of $p$ for which the vectors $(p+1) \hat{i}-3 \hat{j}+p \hat{k}, p \hat{i}+(p+1) \hat{j}-3 \hat{k}$ and $-3 \hat{\mathrm{i}}+\mathrm{p} \hat{\mathrm{j}}+(\mathrm{p}+1) \hat{\mathrm{k}}$ are linearly dependent vectors, are

If $\bar{p}=2 \hat{i}+\hat{k}, \bar{q}=\hat{i}+\hat{j}+\hat{k}, \bar{r}=4 \hat{i}-3 \hat{j}+7 \hat{k}$ and a vector $\overline{\mathrm{m}}$ is such that $\overline{\mathrm{m}} \times \overline{\mathrm{q}}=\overline{\mathrm{r}} \times \overline{\mathrm{q}}, \overline{\mathrm{m}} \cdot \overline{\mathrm{p}}=0$, then $\overline{\mathrm{m}}=\ldots$.

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 $|\overline{\mathrm{a}}+\overline{\mathrm{b}}+\overline{\mathrm{c}}|=\ldots \ldots$.