1. **State the problem:** We are given two vectors $ \vec{OA} = 2\mathbf{i} + 3\mathbf{j} + 4\mathbf{k} $ and $ \vec{OB} = 4\mathbf{i} + 8\mathbf{j} + 5\mathbf{k} $. We want to find the magnitude of the vector $ \vec{AB} $ using the Pythagorean theorem. 2. **Understand the vector $ \vec{AB} $:** The vector $ \vec{AB} $ is the difference between $ \vec{OB} $ and $ \vec{OA} $: $$ \vec{AB} = \vec{OB} - \vec{OA} $$ 3. **Calculate $ \vec{AB} $:** $$ \vec{AB} = (4 - 2)\mathbf{i} + (8 - 3)\mathbf{j} + (5 - 4)\mathbf{k} = 2\mathbf{i} + 5\mathbf{j} + 1\mathbf{k} $$ 4. **Formula for magnitude of a vector:** The magnitude $ |\vec{v}| $ of a vector $ \vec{v} = a\mathbf{i} + b\mathbf{j} + c\mathbf{k} $ is given by the Pythagorean theorem extended to three dimensions: $$ |\vec{v}| = \sqrt{a^2 + b^2 + c^2} $$ 5. **Apply the formula to $ \vec{AB} $:** $$ |\vec{AB}| = \sqrt{2^2 + 5^2 + 1^2} = \sqrt{4 + 25 + 1} = \sqrt{30} $$ 6. **Simplify the magnitude:** $$ |\vec{AB}| = \sqrt{30} \approx 5.477 $$ **Final answer:** The magnitude of $ \vec{AB} $ is $ \sqrt{30} $ or approximately 5.477.