Vector Magnitude Calculator

What this calculator does

The magnitude of a vector is simply its length, how long the arrow is, regardless of which way it points. This finds it from the vector's components, and it works not just in two and three dimensions but on up to five.

Enter the coordinates and it returns the magnitude. It runs right here in the browser.

Using the calculator

1. Choose the space: 2D, 3D, 4D or 5D. The number of coordinate boxes changes to match.
2. Enter the vector's components.
3. Press Calculate.

It returns the magnitude, a single non-negative number. Reset clears the boxes.

What magnitude is

If a vector is an arrow, its magnitude is the length of that arrow, the straight-line distance from its start to its tip. It is written with vertical bars, like |v|, and it is always zero or positive, because a length cannot be negative. Direction plays no part in it: (3, 4) and (-3, -4) point opposite ways but have exactly the same length.

Only one vector has a magnitude of zero, the zero vector (0, 0), which has no length and no direction at all. Every other vector has a positive length.

The formula: Pythagoras, extended

The magnitude is the square root of the sum of the squared components:

|v| = √(x² + y² + z² + ...)

If that looks familiar, it should. In 2D it is exactly the Pythagorean theorem: the vector (x, y) is the hypotenuse of a right triangle with legs x and y, so its length is √(x² + y²). The lovely thing is that the same idea just keeps working. Add a third axis and you add a z² under the root. Add a fourth and fifth, and you add those squares too. Length in any number of dimensions is nothing more than Pythagoras, applied again and again.

Worked examples

In 2D: the vector (3, 4). Its magnitude is √(3² + 4²) = √(9 + 16) = √25 = 5. That is the famous 3-4-5 right triangle showing up again.

In 3D: the vector (2, 3, 6). Its magnitude is √(2² + 3² + 6²) = √(4 + 9 + 36) = √49 = 7. Another clean whole number, though most vectors give a decimal, since most sums of squares are not perfect squares.

Beyond 3D

We can only picture arrows in two or three dimensions, but the maths does not stop there, and neither does this tool. A four or five dimensional vector is just a list of four or five numbers, and its magnitude follows the same rule: square them all, add, take the root. Vectors with many dimensions are everywhere in practice, from a row of data with dozens of features to the state of a physical system, and their length is measured in exactly this way. It is one of those cases where the formula is wiser than our imagination.

Questions people ask

What is the magnitude of a vector?

Its length, the straight-line distance from its start to its tip, found as the square root of the sum of its squared components.

What is the formula for magnitude?

|v| = √(x² + y² + z² + ...), summing the square of every component and taking the square root.

Can a magnitude be negative?

No. A length is always zero or positive. Only the zero vector has a magnitude of zero.

How is it related to the Pythagorean theorem?

In 2D it is the theorem exactly: the vector is the hypotenuse of a right triangle whose legs are its components. Higher dimensions just add more squared terms under the root.

Does it work beyond 3D?

Yes. The same sum-of-squares rule applies in 4D, 5D and beyond, which is why this tool handles up to five dimensions.

References

A note on where this comes from. The magnitude of a vector is its Euclidean norm, the straight-line length given by the square root of the sum of its squared components, a direct generalisation of the Pythagorean theorem to any number of dimensions. For further reading, see Norm (mathematics).

1. The Euclidean norm, |v| = √(x² + y² + ...), the standard measure of a vector's length.
2. The Pythagorean theorem, of which the two-dimensional magnitude is the direct case.