1. The problem states that vectors $\mathbf{a}$ and $\mathbf{b}$ are orthogonal, meaning their dot product is zero: $$\mathbf{a} \cdot \mathbf{b} = 0.$$\n\n2. We are given the magnitudes: $$\|\mathbf{a}\| = \sqrt{2}, \quad \|\mathbf{b}\| = 1.$$\n\n3. We want to find the magnitude of the vector difference: $$\|\mathbf{a} - \mathbf{b}\|.$$\n\n4. Use the formula for the magnitude of a difference of vectors: $$\|\mathbf{a} - \mathbf{b}\|^2 = \|\mathbf{a}\|^2 + \|\mathbf{b}\|^2 - 2(\mathbf{a} \cdot \mathbf{b}).$$\n\n5. Since $\mathbf{a}$ and $\mathbf{b}$ are orthogonal, $\mathbf{a} \cdot \mathbf{b} = 0$, so the formula simplifies to: $$\|\mathbf{a} - \mathbf{b}\|^2 = \|\mathbf{a}\|^2 + \|\mathbf{b}\|^2.$$\n\n6. Substitute the given magnitudes: $$\|\mathbf{a} - \mathbf{b}\|^2 = (\sqrt{2})^2 + 1^2 = 2 + 1 = 3.$$\n\n7. Taking the square root gives: $$\|\mathbf{a} - \mathbf{b}\| = \sqrt{3}.$$\n\n8. Therefore, the correct answer is (a) $$\|\mathbf{a} - \mathbf{b}\| = \sqrt{3}.$$