Given vectors

The two sides of triangle $ABC$ from vertex $A$ are: $\overrightarrow{AB} = 3\hat i + 4\hat k = (3,0,4)$ $\overrightarrow{AC} = 5\hat i - 2\hat j + 4\hat k = (5,-2,4)$

Median through $A$

Let $M$ be the midpoint of side $BC$. Then the median through $A$ is $\overrightarrow{AM}$.

Since position vectors from $A$ to $B$ and $C$ are $\overrightarrow{AB}$ and $\overrightarrow{AC}$, the midpoint vector is: $\overrightarrow{AM} = \frac{\overrightarrow{AB}+\overrightarrow{AC}}{2}$

Compute $\overrightarrow{AM}$

$\overrightarrow{AB}+\overrightarrow{AC} = (3,0,4) + (5,-2,4) = (8,-2,8)$

Therefore, $\overrightarrow{AM} = \frac{1}{2}(8,-2,8) = (4,-1,4)$

Find the length of the median

$|\overrightarrow{AM}| = \sqrt{4^2 + (-1)^2 + 4^2}$ $= \sqrt{16+1+16} = \sqrt{33}$

Match with options

$\sqrt{33}$ corresponds to Option D.

Comparison with stored answer

Stored correct answer: D

Our derived answer: D

Hence, they agree.