Column vector and magnitude

$$\overrightarrow{AB} = \begin{pmatrix} x_2 - x_1 \\ y_2 - y_1 \end{pmatrix} = \mathbf{b} - \mathbf{a} \qquad \left|\overrightarrow{AB}\right| = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$ For $\mathbf{v} = \begin{pmatrix} x \\ y \end{pmatrix}$: $\ |\mathbf{v}| = \sqrt{x^2 + y^2}$.

Operations

$$\begin{pmatrix} a \\ b \end{pmatrix} \pm \begin{pmatrix} c \\ d \end{pmatrix} = \begin{pmatrix} a \pm c \\ b \pm d \end{pmatrix} \qquad k\begin{pmatrix} a \\ b \end{pmatrix} = \begin{pmatrix} ka \\ kb \end{pmatrix}$$

Position vectors

• Midpoint of $AB$: $\ \dfrac{\mathbf{a} + \mathbf{b}}{2}$.
• Point $P$ dividing $AB$ in the ratio $m:n$: $\ \mathbf{p} = \dfrac{n\mathbf{a} + m\mathbf{b}}{m + n}$.

Parallel and collinear

• $\mathbf{a}$ and $\mathbf{b}$ are parallel if $\mathbf{a} = k\mathbf{b}$.
• Points $A$, $B$, $C$ are collinear if $\overrightarrow{AB} = k\,\overrightarrow{BC}$ (or $\overrightarrow{AB}$ is parallel to $\overrightarrow{AC}$).