# Equation of Straight Line in Plane/Point-Slope Form/Parametric Form

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## Theorem

Let $\LL$ be a straight line embedded in a cartesian plane, given in point-slope form as:

$y - y_0 = \paren {x - x_0} \tan \psi$

where $\psi$ is the angle between $\LL$ and the $x$-axis.

Then $\LL$ can be expressed by the parametric equations:

$\begin {cases} x = x_0 + t \cos \psi \\ y = y_0 + t \sin \psi \end {cases}$

## Proof

Let $P_0$ be the point $\tuple {x_0, y_0}$.

Let $P$ be an arbitrary point on $\LL$.

Let $t$ be the distance from $P_0$ to $P$ measured as positive when in the positive $x$ direction.

The equation for $P$ is then:

 $\ds y - y_0$ $=$ $\ds \paren {x - x_0} \tan \psi$ $\ds \leadsto \ \$ $\ds \dfrac {x - x_0} {\cos \psi}$ $=$ $\ds t$ $\ds \dfrac {y - y_0} {\sin \psi}$ $=$ $\ds t$

The result follows.

$\blacksquare$