A line in three dimensional space has vector equation \(\bm{r}(t) = \bm{r}_0 + t\bm{v}.\) The line passing through \(P(x_0, y_0, z_0)\) in the direction \(\bm{u} = \langle u_1, u_2, u_3\rangle\) can be parameterized as \[ x = x_0 + u_1t \qquad y = y_0 + u_2t \qquad z = z_0 + u_3t\,. \]
A plane in space can be defined by a single point and a normal vector \(\bm{n}.\) The vector equation of a plane is then \[\bm{n} \cdot (\bm{r}-\bm{r}_0) = 0\] which, once written out explicitly in terms of components is \(a(x-x_0) + b(y-y_0) + c(z-z_0) = 0\,.\)