Equal, opposite and scaled vectors

Two vectors are equal precisely when they have the same magnitude and direction, wherever the arrows are drawn. Component equality means \(\begin{pmatrix}m\\n\end{pmatrix}=\begin{pmatrix}p\\q\end{pmatrix}\) exactly when \(m=p\) and \(n=q\).

Reversing an arrow negates both components: \(\overrightarrow{BA}=-\overrightarrow{AB}\). Multiplying by a positive scalar keeps direction; a negative scalar reverses it. Length is multiplied by the scalar’s absolute value. The zero vector has no direction.

Example

If \(\mathbf a=\begin{pmatrix}2\\-3\end{pmatrix}\), then \(-\mathbf a=\begin{pmatrix}-2\\3\end{pmatrix}\) and \(3\mathbf a=\begin{pmatrix}6\\-9\end{pmatrix}\). Their magnitudes are \(\sqrt{13}\) and \(3\sqrt{13}\), respectively.

A translation by \(\begin{pmatrix}2\\-3\end{pmatrix}\) moves every point two right and three down. Thus \((1,5)\) maps to \((3,2)\). All sides and angles are preserved because every point moves by the same displacement.