Vector Magnitude Calculator

1. Enter the vector components, separated by commas.
2. Use two for a 2D vector, three for 3D, or more for higher dimensions.
3. Read the magnitude, the length of the vector.
4. See the dimensions and the sum of squares behind it.

Here is what each result means:

The magnitude of a vector is its length, how far it reaches from start to end regardless of direction. For a vector with components, it is the square root of the sum of each component squared. A 2D vector with components 3 and 4 has a magnitude of 5, since the square root of 9 plus 16 is 5.

Magnitude turns a set of components into a single length. It answers how fast, how far or how strong, while the direction is described separately. The idea extends to any number of dimensions, using the same sum-of-squares rule, and it is a direct extension of the Pythagorean theorem.

Because each component is squared before adding, the sign of a component never matters, and a longer component in any direction always adds to the total length. A vector and its reverse, which point opposite ways, share the exact same magnitude. Only the zero vector, with every component zero, has a magnitude of zero.

It squares each component, adds them, and takes the square root.

1. Square each component of the vector.
2. Add all the squares together.
3. Take the square root of that total for the magnitude.

In two dimensions this is exactly the Pythagorean theorem, and it matches the straight-line distance from the origin.

Find the magnitude of the 3D vector with components 1, 2, 2.

Calculation: |v| = sqrt(1^2 + 2^2 + 2^2) = sqrt(1 + 4 + 4) = sqrt(9) = 3. The vector has length 3. Its unit vector, the same direction but length 1, is about (0.333, 0.667, 0.667).

The same rule applies whatever the dimension; you just add more squared terms.

These clean results come from Pythagorean triples; most vectors give a magnitude that is not a whole number.

A vector has two parts, and magnitude is only one of them.

Dividing a vector by its magnitude gives the unit vector, which keeps the direction but sets the length to one.

The Size of the Components

Larger components give a longer vector, and each one adds through its square.

The Number of Dimensions

More components can add more length, though a component of zero adds nothing.

Sign Does Not Matter

Squaring removes the sign, so a component of minus 4 adds the same as plus 4.

Physics

Find the speed from a velocity vector, or the size of a force from its components.

Geometry

Measure the length of a vector or the distance of a point from the origin.

Graphics and Games

Normalize a vector to a unit length for directions and movement.

1. Adding Components Without Squaring

Magnitude squares each component first. Adding the raw values gives the wrong length.

2. Forgetting the Square Root

The sum of squares is the magnitude squared. Take the square root for the length.

3. Worrying About Signs

Squaring makes every component positive, so a negative component is fine.

4. Confusing Magnitude with a Component

The magnitude is the whole length, not any single component.

5. Mismatched Dimensions

All components must belong to the same vector. Do not mix vectors of different lengths.

The formula is exact; only the displayed decimals are rounded.

It is the length of the vector, how far it reaches regardless of direction. It equals the square root of the sum of the squared components, so the vector (3, 4) has a magnitude of 5.

Square each component, add the squares, and take the square root. For (1, 2, 2), that is the square root of 1 plus 4 plus 4, which is the square root of 9, or 3.

Yes. The same rule applies in any number of dimensions. You simply add one squared term for each extra component before taking the square root.

In two dimensions, yes. The magnitude of (a, b) is the square root of a squared plus b squared, which is the hypotenuse of a right triangle with legs a and b.

It is a vector with a magnitude of 1 that points in the same direction. You get it by dividing each component by the magnitude, which keeps the direction but sets the length to one.

Not for magnitude. Squaring each component removes its sign, so a component of minus 4 contributes the same as plus 4 to the length.

Zero. Every component is zero, so the sum of squares is zero and its square root is zero. The zero vector has no defined direction.

It turns a vector into a single size: the magnitude of a velocity is speed, and the magnitude of a force is its strength. The direction is described separately.

No. The calculation runs in your browser and nothing you enter is stored or sent anywhere, unless you choose Save, which keeps the result only on this device.

• Magnitude (mathematics) (Wikipedia).
• Vector magnitude explained (Maths Is Fun).
• Unit vector (Wikipedia).