Magnitude

A vector's length, $|\mathbf{v}|=\sqrt{v_1^2+v_2^2+v_3^2}$ — the distance formula from the origin.

The magnitude (or norm) of a vector is its length, computed as the distance formula applied from the origin to the vector's tip: $|\mathbf{v}|=\sqrt{v_1^2+v_2^2+v_3^2}$ in space (or $\sqrt{v_1^2+v_2^2}$ in the plane). Scaling a vector by $k$ scales its magnitude by $|k|$ , and reverses its direction exactly when $k<0$ .

Examples

For $\mathbf{v}=\langle 3,4\rangle$ , $|\mathbf{v}|=\sqrt{3^2+4^2}=5$ .
The distance from $P_1=(2,-1,5)$ to $P_2=(-3,4,1)$ is $|\overrightarrow{P_1P_2}|=\sqrt{66}\approx 8.12$ .