Adding routes and subtracting vectors

Join vectors head to tail: travelling \(A\to B\to C\) has the same net displacement as \(A\to C\). Thus \(\overrightarrow{AB}+\overrightarrow{BC}=\overrightarrow{AC}\). Add or subtract column vectors component by component.

Example

If \(\mathbf a=\begin{pmatrix}5\\0\end{pmatrix}\) and \(\mathbf b=\begin{pmatrix}1\\4\end{pmatrix}\), then \(\mathbf a+\mathbf b=\begin{pmatrix}6\\4\end{pmatrix}\) and \(\mathbf a-\mathbf b=\begin{pmatrix}4\\-4\end{pmatrix}\). Subtraction adds the opposite vector.

In a parallelogram \(OABC\) with consecutive vertices, \(\overrightarrow{OA}=\mathbf a\), \(\overrightarrow{OC}=\mathbf b\), the diagonal \(\overrightarrow{OB}=\mathbf a+\mathbf b\). Opposite sides have equal vectors only when their arrows point the same way.