Math Vectors Vector length (or magnitude)

# Vector length (or magnitude)

For determining the vector length (or magnitude) of a vector $|\vec{v}|$ we use the Pythagorean theorem.

The vector length (or magnitude) of $\vec{v}=\begin{pmatrix}v_1\\v_2\end{pmatrix}$ is:

$|\vec{v}|=\left|\begin{pmatrix}v_1\\v_2\end{pmatrix}\right|=\sqrt{v_1^2+v_2^2}$

In three-dimensional space:

$|\vec{v}|=\left|\begin{pmatrix}v_1\\v_2\\v_3\end{pmatrix}\right|=\sqrt{v_1^2+v_2^2+v_3^2}$
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### Remember

The vector length (or magnitude) is the length of its arrow and corresponds to the distance between initial point and terminal point.

For determining the length of the arrow (and thus the magnitude of the vector), think of the following triangle. Using the Pythagorean theorem you will find the length of the arrow.

### Examples

Determine the vector length $\vec{a}=\begin{pmatrix}3\\4\end{pmatrix}$

$|\vec{a}|=\sqrt{3^2+4^2}=\sqrt{25}=5$

Determine the vector length $\vec{a}=\begin{pmatrix}2\\3\\6\end{pmatrix}$

$|\vec{a}|=\sqrt{2^2+3^2+6^2}=\sqrt{49}$ $=7$