A worked example: the vector (3, 4)
Subtract (0, 0) from (3, 4) and the result is unchanged: (3, 4), with a magnitude of 5. That 3-4-5 result isn't a coincidence — it's the classic Pythagorean triple, because magnitude is computed as the square root of the sum of squared components: √(3² + 4²) = √(9 + 16) = √25 = 5. A 2D vector's magnitude is just the length of the hypotenuse of the right triangle formed by its components.
The formula: square, sum, then root
For any vector, take each component, square it, add all the squares together, and take the square root of that sum. The vector (6, 8) demonstrates the same pattern scaled up: √(6² + 8²) = √(36 + 64) = √100 = 10. Every extra dimension just adds one more squared term under the root — the formula doesn't change shape, it just keeps summing.
A three-dimensional example: (1,2,3) + (4,5,6)
Adding (1, 2, 3) and (4, 5, 6) component-by-component gives (5, 7, 9). Its magnitude is √(5² + 7² + 9²) = √(25 + 49 + 81) = √155 ≈ 12.4499 — the same square-root-of-sum-of-squares rule, just with a third term added under the root. Nothing about moving from 2D to 3D changes the underlying formula.
Why the same formula scales to any dimension
Because magnitude only ever sums squared components before taking one square root, it doesn't matter whether a vector has two components, three, or twenty — the formula reads the same way. That generality is also what makes it interesting to shrink a vector down to a single dimension: with only one component, the sum-of-squares-then-root formula reduces to something much simpler, as the companion article on why a vector's magnitude reduces to absolute value in one dimension works out numerically.