Vectors & Transformations

Vectors

A vector has magnitude and direction, written as a column vector $\begin{pmatrix} x \\ y \end{pmatrix}$ ($x$ across, $y$ up). Add or subtract by combining components; multiply by a scalar component-wise:

$$\begin{pmatrix} 3 \\ 1 \end{pmatrix} + \begin{pmatrix} -1 \\ 4 \end{pmatrix} = \begin{pmatrix} 2 \\ 5 \end{pmatrix}, \qquad 2\begin{pmatrix} 3 \\ -2 \end{pmatrix} = \begin{pmatrix} 6 \\ -4 \end{pmatrix}$$

The magnitude (length) of $\begin{pmatrix} x \\ y \end{pmatrix}$ is $\sqrt{x^2 + y^2}$. In geometry, $\overrightarrow{AB} = \mathbf{b} - \mathbf{a}$. Parallel vectors are scalar multiples of each other.