Directed displacement and magnitude

A vector has magnitude and direction. The vector \(\overrightarrow{AB}\) describes displacement from \(A\) to \(B\); its magnitude \(|\overrightarrow{AB}|\) is a length. A column vector \(\begin{pmatrix}p\\q\end{pmatrix}\) means \(p\) units horizontally and \(q\) units vertically, with right/up positive.

Example

From \(A=(1,2)\) to \(B=(4,6)\), move three units right and four up: \(\overrightarrow{AB}=\begin{pmatrix}3\\4\end{pmatrix}\).

By Pythagoras, \(\left|\begin{pmatrix}p\\q\end{pmatrix}\right|=\sqrt{p^2+q^2}\). Here the magnitude is \(\sqrt{3^2+4^2}=5\) units.

The vector \(\begin{pmatrix}-3\\4\end{pmatrix}\) also has magnitude \(5\), but points in a different direction. Equal length alone does not make equal vectors.