Directional and orthogonal form of a line

Directional and orthogonal form of a line

Directional form:

\begin{align*} \begin{pmatrix} x_1 \\ x_2 \end{pmatrix} &= \begin{pmatrix} a_1 \\ a_2 \end{pmatrix} + \alpha \begin{pmatrix} d_1 \\ d_2 \end{pmatrix} \end{align*}

or, concisely

\begin{align} x = a + \alpha d \end{align}

Let $v$ be a vector perpendicular to $d$. Transpose both sides and multiply by $v$:

\begin{align} x'v &= a'v + \alpha d'v = a'v = c \end{align}

Where c is a constant scalar. This is called the orthogonal form.

If you see it as $f(x) = x'v $, then $v$ is the gradient of $f$.