Canonical equation of a line in space: horizontal and vertical lines

Question

I have a question about canonical equation of a line in 3d space: how can I handle vertical and horizontal lines? One of direction vector's values will be just $0$, but this will mess up the equation, because I can't put $0$ value in denominator. Can I just describe vertical line using only $x$ and $z$ coordinates of its points?

Answer 1

You will sometimes find the conventional notation

$$\frac{x-x_0}l=\frac{y-y_0}m=\frac{z-z_0}n$$

with a zero allowed at the denominator, like

$$\frac{x-x_0}l=\frac{y-y_0}0=\frac{z-z_0}n,$$

which must be interpreted as $y=y_0$.

You cannot drop this condition, as

$$\frac{x-x_0}l=\frac{z-z_0}n$$

alone is the equation of a plane (parallel to the $y$ axis).

The parametric equation avoids the singular cases while using exactly the same parameters, so it can be preferred.