Step-by-Step Solution

1. Let the position vectors of A, B, and C be \(\vec{a}\), \(\vec{b}\), and \(\vec{c}\) respectively.

   Given: \(\vec{AB} = \hat{j} + \hat{k}\) and \(\vec{AC} = 3\hat{i} - \hat{j} + 4\hat{k}\)

2. Let M be the midpoint of BC. Then AM is the median.

   \(\vec{AM} = \vec{m} - \vec{a} = \frac{\vec{b} + \vec{c}}{2} - \vec{a} = \frac{1}{2}(\vec{b} - \vec{a} + \vec{c} - \vec{a}) = \frac{1}{2}(\vec{AB} + \vec{AC})\)

3. Substitute the given vectors:

   \(\vec{AM} = \frac{1}{2}[(\hat{j} + \hat{k}) + (3\hat{i} - \hat{j} + 4\hat{k})] = \frac{1}{2}(3\hat{i} + 5\hat{k})\)

4. Find the magnitude of \(\vec{AM}\):

   \(|\vec{AM}| = \frac{1}{2}\sqrt{3^2 + 0^2 + 5^2} = \frac{1}{2}\sqrt{9 + 25} = \frac{1}{2}\sqrt{34}\)

5. Therefore, the length of the median through A on BC is \(\frac{\sqrt{34}}{2}\) units.