Whilst it can be useful to express the equation of a line in its vector form $latex \mathbf{r} = \mathbf{a} + t \mathbf{p} $, it is also important that we are able to describe the line in a cartesian form, without the parameter t.

The expansion of the vector form gives the following general equation:
\left( \begin{array}{c} x \\ y \\ z \end{array} \right) = \left( \begin{array}{c} a \\ b \\ c \end{array} \right) + t \left( \begin{array}{c} l \\ m \\ n \end{array} \right)

If we rearrange these three equations for t we get:
t = \frac{x - a}{l} \\ t = \frac {y - b} {m} \\ t = \frac {z - c} {n}

Setting all these equal, as t is common to the equations, we get the following:

\frac {x - a}{l} = \frac {y - b} {m} = \frac {z - c} {n}