How this instrument works
A vector's magnitude is simply its length: how far its tip sits from its tail, regardless of which way it happens to point. Because a vector's x-component and y-component are exactly the two legs of a right triangle — one running horizontally, the other vertically, meeting at a right angle — the magnitude is found with a direct application of the Pythagorean theorem: |v| = √(x² + y²), the same relationship that gives a hypotenuse's length from a right triangle's two legs.
This is the plainer, purely numerical half of a vector's full description; the other half is its DIRECTION, which angle it points at, a separate question this calculator does not answer (the companion vector-direction page on this site handles that half). Magnitude answers 'how far,' direction answers 'which way,' and a vector genuinely needs both to be fully pinned down.
Magnitude is always non-negative, and it equals zero only for the zero vector, (0, 0) — a vector that hasn't gone anywhere at all. Scaling every component of a vector by the same factor scales the magnitude by that same factor exactly, since the factor pulls straight out of the square root: doubling both x and y doubles the magnitude, not quadruples it, even though each component is individually squared inside the formula.
- Enter the vector's horizontal component into the x field.
- Enter the vector's vertical component into the y field.
- Read Magnitude: the sheet computes √(x² + y²) directly.
- For the vector's direction as well as its length, use the companion Vector Direction Calculator on this site.
Worked example — the length of (3, 4)
The vector (3, 4) has a magnitude of |v| = √(3² + 4²) = √(9 + 16) = √25 = 5 — a direct 3-4-5 Pythagorean triple, with the vector's own components standing in for a right triangle's two legs and the magnitude standing in for its hypotenuse.
Compare the vector (5, 12): |v| = √(25 + 144) = √169 = 13, another classic Pythagorean triple, this time 5-12-13. The zero vector, (0, 0), sits at the opposite extreme: |v| = √(0 + 0) = 0, the only vector with no length at all.
Questions
What is the formula for a vector's magnitude?
|v| = √(x² + y²), where x and y are the vector's components. This is a direct application of the Pythagorean theorem, treating the vector's two components as the legs of a right triangle and its magnitude as that triangle's hypotenuse.
Can a vector's magnitude be negative?
No — magnitude is always zero or positive, since it comes from a square root, which by convention always returns the non-negative root. A vector can point in a negative direction along an axis, but its overall length is never itself a negative number.
What is the magnitude of the zero vector?
Exactly 0 — the only vector with no length at all, sitting precisely at the origin with no displacement in either direction.
How does scaling a vector affect its magnitude?
Multiplying both components by the same factor multiplies the magnitude by that exact same factor — doubling a vector's components doubles its length, tripling them triples its length, and so on, since the scale factor pulls cleanly out of the square root in the magnitude formula.
Is vector magnitude the same as vector direction?
No — magnitude answers 'how long is this vector,' while direction answers 'which way does it point,' a completely separate question handled by this site's companion Vector Direction Calculator. A full description of a vector needs both numbers together.