The vectors $\overline{AB} =3\overline{i}+4\overline{k}$ and $\overline{AC} =5\overline{i}-2\overline{ j}+4\overline{k}$ are the sides of a triangle ABC. The length of the median through A is

$$\begin{gathered} \overline{\mathrm{AC}}=\overline{\mathrm{AB}}+\overline{\mathrm{BC}} \Rightarrow \overline{\mathrm{BC}}=5\mathrm{i}-2\mathrm{j}+4\mathrm{k}-3\mathrm{i}-4\mathrm{k} \\ =2\mathrm{i}-2\mathrm{j} \\ \mathrm{a}=\left|\overline{\mathrm{BC}}\right|=\sqrt{4+4}=\sqrt{8} \mathrm{b}=\left|\overline{\mathrm{CA}}\right|=\sqrt{25+4+16}=\sqrt{45} \\ \mathrm{C}=\left|\overline{\mathrm{AB}}\right|=\sqrt{9+16}=\sqrt{25} \\ \mathrm{length} \mathrm{of} \mathrm{median} \mathrm{through} \mathrm{A}=\sqrt{\frac{2{\mathrm{b}}^2+2{\mathrm{c}}^2-{\mathrm{a}}^2}{4}} \\ =\sqrt{\frac{90+50-8}{4}} =\sqrt{33} \end{gathered}$$