To solve this, let's start with the given vector equation:

$$\vec{c} = \vec{a} - \vec{b}$$

Given vectors are:

$$\overrightarrow{\mathrm{a}} = \alpha \hat{i} + 4 \hat{j} + 2 \hat{k}$$

$$\overrightarrow{\mathrm{b}} = 5 \hat{i} + 3 \hat{j} + 4 \hat{k}$$

Then, the vector $$\overrightarrow{\mathrm{c}}$$ is:

$$\overrightarrow{\mathrm{c}} = (\alpha \hat{i} + 4 \hat{j} + 2 \hat{k}) - (5 \hat{i} + 3 \hat{j} + 4 \hat{k})$$

$$\overrightarrow{\mathrm{c}} = (\alpha - 5) \hat{i} + (4 - 3) \hat{j} + (2 - 4) \hat{k}$$

$$\overrightarrow{\mathrm{c}} = (\alpha - 5) \hat{i} + 1 \hat{j} - 2 \hat{k}$$

The area of the triangle formed by vectors $$\vec{a}$$ and $$\vec{b}$$ is given by the magnitude of the cross product of $$\vec{a}$$ and $$\vec{b}$$, divided by 2:

$$\text{Area} = \frac{1}{2} |\vec{a} \times \vec{b}| = 5 \sqrt{6}$$

This implies:

$$|\vec{a} \times \vec{b}| = 10 \sqrt{6}$$

Let's find $$\vec{a} \times \vec{b}$$:

$$\vec{a} \times \vec{b} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ \alpha & 4 & 2 \\ 5 & 3 & 4 \end{vmatrix} = \hat{i}(4 \cdot 4 - 2 \cdot 3) - \hat{j}(\alpha \cdot 4 - 2 \cdot 5) + \hat{k}(\alpha \cdot 3 - 4 \cdot 5)$$

$$\vec{a} \times \vec{b} = \hat{i}(16 - 6) - \hat{j}(4\alpha - 10) + \hat{k}(3\alpha - 20)$$

$$\vec{a} \times \vec{b} = \hat{i}(10) - \hat{j}(4\alpha - 10) + \hat{k}(3\alpha - 20)$$

Magnitude of $$\vec{a} \times \vec{b}$$:

$$|\vec{a} \times \vec{b}| = \sqrt{10^2 + (4\alpha - 10)^2 + (3\alpha - 20)^2}$$

We know:

$$\sqrt{10^2 + (4\alpha - 10)^2 + (3\alpha - 20)^2} = 10 \sqrt{6}$$

Squaring both sides:

$$100 + (4\alpha - 10)^2 + (3\alpha - 20)^2 = 600$$

$$100 + 16\alpha^2 - 80\alpha + 100 + 9\alpha^2 - 120\alpha + 400 = 600$$

$$25\alpha^2 - 200\alpha + 600 = 600$$

$$25\alpha^2 - 200\alpha = 0$$

$$\alpha^2 - 8\alpha = 0$$

$$\alpha(\alpha - 8) = 0$$

Since $$\alpha$$ is a positive real number:

$$\alpha = 8$$

Then, the vector $$\vec{c}$$ is:

$$\vec{c} = (8 - 5)\hat{i} + 1\hat{j} - 2\hat{k}$$

$$\vec{c} = 3\hat{i} + 1\hat{j} - 2\hat{k}$$

Magnitude squared of $$\vec{c}$$ is:

$$|\vec{c}|^2 = 3^2 + 1^2 + (-2)^2$$

$$|\vec{c}|^2 = 9 + 1 + 4$$

$$|\vec{c}|^2 = 14$$

Therefore, the correct option is:

Option A: 14