The points $A$, $B$, and $C$ are collinear. $B$ divides $AC$ in the ratio $2:3$.

Step 1: Find position vectors

Position vector of $A = \hat{i} - 2\hat{j} - 8\hat{k}$

Position vector of $B = 5\hat{i} + 0\hat{j} - 2\hat{k}$

Position vector of $C = 11\hat{i} + 3\hat{j} + 7\hat{k}$

Step 2: Find vectors AB, BC, and AC

$$\vec{AB} = (5-1)\hat{i} + (0+2)\hat{j} + (-2+8)\hat{k} = 4\hat{i} + 2\hat{j} + 6\hat{k}$$ $$\vec{BC} = (11-5)\hat{i} + (3-0)\hat{j} + (7+2)\hat{k} = 6\hat{i} + 3\hat{j} + 9\hat{k}$$ $$\vec{AC} = (11-1)\hat{i} + (3+2)\hat{j} + (7+8)\hat{k} = 10\hat{i} + 5\hat{j} + 15\hat{k}$$

Step 3: Calculate magnitudes

$$|\vec{AB}| = \sqrt{16+4+36} = \sqrt{56} = 2\sqrt{14}$$ $$|\vec{BC}| = \sqrt{36+9+81} = \sqrt{126} = 3\sqrt{14}$$ $$|\vec{AC}| = \sqrt{100+25+225} = \sqrt{350} = 5\sqrt{14}$$

Step 4: Verify collinearity

$$|\vec{AC}| = 5\sqrt{14} = 2\sqrt{14} + 3\sqrt{14} = |\vec{AB}| + |\vec{BC}|$$

Since $|\vec{AC}| = |\vec{AB}| + |\vec{BC}|$, the points $A$, $B$, and $C$ are collinear.

Step 5: Find the ratio

$$|\vec{AB}| : |\vec{BC}| = 2\sqrt{14} : 3\sqrt{14} = \textbf{2:3}$$

∴ $B$ divides $AC$ in the ratio 2:3.