1. The problem is to find the magnitude of the vector $\mathbf{v} = 5\mathbf{i} - 4\mathbf{j} + 2\mathbf{k}$.\n\n2. The formula for the magnitude of a vector $\mathbf{v} = a\mathbf{i} + b\mathbf{j} + c\mathbf{k}$ is given by:\n$$\|\mathbf{v}\| = \sqrt{a^2 + b^2 + c^2}$$\nThis formula comes from the Pythagorean theorem extended to three dimensions.\n\n3. Substitute the components of the vector into the formula:\n$$\|\mathbf{v}\| = \sqrt{5^2 + (-4)^2 + 2^2}$$\n\n4. Calculate the squares:\n$$\|\mathbf{v}\| = \sqrt{25 + 16 + 4}$$\n\n5. Sum the values inside the square root:\n$$\|\mathbf{v}\| = \sqrt{45}$$\n\n6. Simplify the square root if possible. Since $45 = 9 \times 5$, we have:\n$$\|\mathbf{v}\| = \sqrt{9 \times 5} = \sqrt{9} \times \sqrt{5} = 3\sqrt{5}$$\n\n7. Therefore, the magnitude of the vector $5\mathbf{i} - 4\mathbf{j} + 2\mathbf{k}$ is $3\sqrt{5}$.