Given : $\overrightarrow {AB}=3\hat i +4\hat k\:\:\:and\:\:\:\overrightarrow {AC}=5\hat i-2\hat j+4\hat k$
$\overrightarrow {AB}+\overrightarrow {AC}=8\hat i-2\hat j+8\hat k$
Median through $A$ of a $\Delta \:ABC$ is $\large\frac{\overrightarrow {AB}+\overrightarrow {AC}}{2}$
$=\large\frac{1}{2}$$(8\hat i-2\hat j+8\hat k)=4\hat i-\hat j+4\hat k$
$\therefore$ length of median = $\sqrt {16+1+16}=\sqrt {33}$