The position vectors of the vertices of $∆\mathrm{ABC}$ are $3\overline{\mathrm{i}}+4\overline{\mathrm{j}}-\mathrm{k}, \overline{\mathrm{i}}+3\overline{\mathrm{j}}+\overline{\mathrm{k}},5(\overline{\mathrm{i}}+\overline{\mathrm{j}}+\overline{\mathrm{k}})$ respectively. The magnitude of the altitude from A onto the side BC is

$$\begin{gathered} \overline{\mathrm{AB}}=-2\mathrm{i}-\mathrm{j}+2\mathrm{k} \\ \overline{\mathrm{AC}}=2\mathrm{i}+\mathrm{j}+6\mathrm{k} \\ \overline{\mathrm{AB}}\times \overline{\mathrm{AC}}=\left|\begin{array}{ccc}\mathrm{i}& \mathrm{j}& \mathrm{k}\\ -2& -1& 2\\ 2& 1& 6\end{array}\right|=\mathrm{i}(-8)-\mathrm{j}(-16)+\mathrm{k}\left(0\right) \\ =8(-\mathrm{i}+2\mathrm{j}) \\ \mathrm{area}=\frac{1}{2}|\overline{\mathrm{AB}}\times \overline{\mathrm{AC}}|=\frac{1}{2} 8\sqrt{1+4}=4\sqrt{5} \\ \left|\overline{\mathrm{BC}}\right|=\sqrt{16+4+16}=6 \\ 4\sqrt{5}=\frac{1}{2}\times 6\times \mathrm{p} \Rightarrow \mathrm{p}=\frac{4\sqrt{5}}{3} \end{gathered}$$