vector equation of a line in $\mathbb{R}^2$

Question

How do I get the vector equation of the following line, I can't figure it out:

$y = 0$, $x \in [0,20]$.

$y = x - 20$, $x > 20$.

So the equation passes through all the points $(18,0), (19,0), (25,5)$ for example.

Where $y \ge 0$ all times.

Thanks.

Answer 1

Consider the Heaviside function:

$$ \Theta(x) = \left\{ \begin{array}{rl} 0, & x < 0 \\ 1, & x \geq 0 \end{array} \right. $$

then

$$ \boldsymbol{r}(t) = \begin{pmatrix} x(t) \\ y(t) \end{pmatrix} = \begin{pmatrix} t \\ (t-20)\Theta(t-20) \end{pmatrix}, t \geq 0 $$