If the vectors $$\vec{AB} = 3\hat{i} + 4\hat{k}$$ and $$\vec{AC} = 5\hat{i} - 2\hat{j} + 4\hat{k}$$ are the sides of a triangle $$ABC$$, then the length of the median through $$A$$ is:

We have the two side vectors of the triangle taken from vertex $$A$$:

$$\vec{AB} = 3\hat{i} + 4\hat{k}, \qquad \vec{AC} = 5\hat{i} - 2\hat{j} + 4\hat{k}.$$

The median from a vertex goes to the midpoint of the opposite side. In vector language, the midpoint $$M$$ of the side $$BC$$ is reached by taking the average of the position vectors of $$B$$ and $$C$$ with respect to $$A$$. Hence the vector along the median $$\vec{AM}$$ is one-half of the sum of the two side vectors. Stating the formula,

$$\vec{AM} = \dfrac{1}{2}\left(\vec{AB} + \vec{AC}\right).$$

Now we add the two given vectors term by term:

$$\vec{AB} + \vec{AC} = \bigl(3\hat{i} + 4\hat{k}\bigr) + \bigl(5\hat{i} - 2\hat{j} + 4\hat{k}\bigr) = (3+5)\hat{i} + (0-2)\hat{j} + (4+4)\hat{k}.$$

Simplifying each component we get

$$\vec{AB} + \vec{AC} = 8\hat{i} - 2\hat{j} + 8\hat{k}.$$

Substituting this result in the formula for the median,

$$\vec{AM} = \dfrac{1}{2}\left(8\hat{i} - 2\hat{j} + 8\hat{k}\right) = 4\hat{i} - \hat{j} + 4\hat{k}.$$

To find the length of the median, we need the magnitude of $$\vec{AM}$$. For any vector $$\vec{v}=a\hat{i}+b\hat{j}+c\hat{k}$$, the magnitude formula is

$$|\vec{v}| = \sqrt{a^{2}+b^{2}+c^{2}}.$$

Applying this to $$\vec{AM}=4\hat{i}-\hat{j}+4\hat{k}$$, we calculate

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

Thus the length of the median through vertex $$A$$ is $$\sqrt{33}$$.

Hence, the correct answer is Option A.