Step 1: Set up the cube and identify its face diagonals.
Place a unit cube with one corner at the origin. Three adjacent faces meeting at the origin have diagonals along directions that connect opposite corners of each face.
Step 2: Identify the three diagonal directions.
The three face diagonals (from the common vertex along diagonals of the three adjacent faces) have directions: $\vec{d_1} = \hat{i}+\hat{j}$, $\vec{d_2} = \hat{j}+\hat{k}$, $\vec{d_3} = \hat{i}+\hat{k}$. Each has magnitude $\sqrt{2}$.
Step 3: Write unit vectors along each diagonal.
$\hat{d_1} = \frac{\hat{i}+\hat{j}}{\sqrt{2}}$, $\hat{d_2} = \frac{\hat{j}+\hat{k}}{\sqrt{2}}$, $\hat{d_3} = \frac{\hat{i}+\hat{k}}{\sqrt{2}}$.
Step 4: Write the three vectors with given magnitudes $a, 2a, 3a$.
$\vec{v_1} = a \cdot \frac{\hat{i}+\hat{j}}{\sqrt{2}}$, $\vec{v_2} = 2a \cdot \frac{\hat{j}+\hat{k}}{\sqrt{2}}$, $\vec{v_3} = 3a \cdot \frac{\hat{i}+\hat{k}}{\sqrt{2}}$.
Step 5: Find the sum vector.
$\vec{v_1}+\vec{v_2}+\vec{v_3} = \frac{a}{\sqrt{2}}[(1+3)\hat{i} + (1+2)\hat{j} + (2+3)\hat{k}] = \frac{a}{\sqrt{2}}[4\hat{i}+3\hat{j}+5\hat{k}]$.
Magnitude $= \frac{a}{\sqrt{2}} \cdot \sqrt{16+9+25} = \frac{a}{\sqrt{2}} \cdot \sqrt{50} = \frac{a \cdot 5\sqrt{2}}{\sqrt{2}} = 5a$.
Step 6: Match with options.
The magnitude is $5a$, which is option (2).
\[ \boxed{5a} \]