Continuous equation of a straight line in the space

If the parametric equations $v_{1}, v_{2}$ and $v_{3}$ are different from $0$ , we can isolate the parameter $k$ in all $3$ equations: $k = \frac{x - a_{1}}{v_{1}} k = \frac{y - a_{2}}{v_{2}} k = \frac{z - a_{3}}{v_{3}}$ In equating the obtained expressions, we have: $\frac{x - a_{1}}{v_{1}} = \frac{y - a_{2}}{v_{2}} = \frac{z - a_{3}}{v_{3}}$ which are the continuous equations of the straight line.

Example

Parametric equations of the straight line going through point $A = (- 1, 1, 3)$ with $\vec{v} = (3, - 2, 1)$ as director vector are: $\begin{array}{rcl} x & = & - 1 + 3 k \\ y & = & 1 - 2 k \\ z & = & 3 + k \end{array}}$

Isolating $k$ and by equating: $\frac{x + 1}{3} = \frac{y - 1}{- 2} = z - 3$ which are the continuous equations of the straight line.