### Magnitude of a vector definition

The magnitude of a vector is the length of the vector. The magnitude of the vector $\vc{a}$ is denoted as $\| \vc{a} \|$. See the introduction to vectors for more about the magnitude of a vector.

Formulas for the magnitude of vectors in two and three dimensions in terms of their coordinates are derived in this page. For a two-dimensional vector $\vc{a}=(a_1,a_2)$, the formula for its magnitude is \begin{gather*} \| \vc{a} \| = \sqrt{a_1^2+a_2^2}. \end{gather*} For a three-dimensional vector $\vc{a}=(a_1,a_2,a_3)$, the formula for its magnitude is \begin{gather*} \| \vc{a} \| = \sqrt{a_1^2+a_2^2 + a_3^2}. \end{gather*}

The formula for the magnitude of a vector can be generalized to arbitrary dimensions. For example, if $\vc{a} = (a_1, a_2, a_3, a_4)$ is a four-dimensional vector, the formula for its magnitude is \begin{gather*} \| \vc{a} \| = \sqrt{a_1^2+a_2^2 + a_3^2 + a_4^2}. \end{gather*}