1. **State the problem:** Find the norm (or magnitude) of the vector $\mathbf{v} = (-1, 2, 4)$.\n\n2. **Formula:** The norm of a vector $\mathbf{v} = (v_1, v_2, v_3)$ in 3D space is given by the formula: $$\|\mathbf{v}\| = \sqrt{v_1^2 + v_2^2 + v_3^2}.$$\n\n3. **Apply the formula:** Substitute the components of $\mathbf{v}$ into the formula: $$\|\mathbf{v}\| = \sqrt{(-1)^2 + 2^2 + 4^2}.$$\n\n4. **Calculate each term:** $$(-1)^2 = 1, \quad 2^2 = 4, \quad 4^2 = 16.$$\n\n5. **Sum the squares:** $$1 + 4 + 16 = 21.$$\n\n6. **Take the square root:** $$\|\mathbf{v}\| = \sqrt{21}.$$\n\n7. **Interpretation:** The norm of the vector $\mathbf{v}$ is $\sqrt{21}$, which is approximately 4.58. This represents the length or magnitude of the vector in 3D space.