# Magnitude of a Vector

This online calculator calculates the magnitude of a vector

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This online calculator calculates the magnitude of a vector, either a **free vector** using its coordinates or a **bound vector** using coordinates of its initial and terminal points. Some recap theory and formulas can be found below the calculator.

### The magnitude of a vector

Here we talk about Euclidean vector, which is a geometric object that has magnitude (or length) and direction. Graphically it can be represented as an arrow, connecting an *initial point* with a *terminal point*. Such vector is called **bound vector**. It is defined by initial point and terminal point coordinates. When you care only about the magnitude and the direction of the vector, and not about the particular initial point, such vector is called a **free vector**. Free vector is equivalent to the bound vector whose initial point is the origin.

The **length** or **magnitude** or **norm** of the vector a is denoted by **‖a‖** or, less commonly, **|a|**, which is not to be confused with the absolute value (a scalar "norm").

The length of the vector a can be computed with the Euclidean norm

which is a consequence of the Pythagorean theorem since the basis vectors e1, e2, e3 are orthogonal unit vectors.^{1}

In case of three-dimensional space with *x*, *y* and *z* coordinates the formula becomes

for free vector

and

for bound vector

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