Magnitude of a Vector

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. You can find theory and formulas below the calculator.

Magnitude of a Vector

Vector type

Free vector

Bound vector

Vector coordinates

x

y

z

Initial point coordinates

x

y

z

Terminal point coordinates

x

y

z

Calculation precision

Digits after the decimal point: 2

Calculate

Magnitude

 

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The magnitude of a vector

Here we talk about the Euclidean vector, a geometric object with 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 an 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. The 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").

You can compute the length of the vector 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