Lines in three-dimensional space can be described by using parametrization. The parametric equation of a line through $\left(x_0,y_0,z_0\right)$ in the direction of the vector $\overrightarrow{v}=\left(a,b,c\right)$ is written like this:

This can also expressed as a vector in this way:

In three dimensions, we need two equations to describe a line. They’re written like this:

$$\frac{x-x_0}{a}=\frac{y-y_0}{b}=\frac{z-z_0}{c}$$

This is another way of writing the two equations

$$\frac{x-x_0}{a}=\frac{y-y_0}{b}$$

and

$$\frac{y-y_0}{b}=\frac{z-z_0}{c}.$$

When you have a parametric equation of a line, you can find these equations by rearranging the coordinate form of the parametric equation to all be expressions for $t$. Then you can set these expressions equal to each other to get the equations above.