Parametric equations of a plane

If we separate the vector equation component by component we obtain ${\begin{array}{rcl} x & = & a_{1} + λ \cdot v_{1} + μ \cdot w_{1} \\ y & = & a_{2} + λ \cdot v_{2} + μ \cdot w_{2} \\ z & = & a_{3} + λ \cdot v_{3} + μ \cdot w_{3} \end{array}$

which is precisely the parametric equations of the plane.

Example

Consider points $A = (1, - 3, 5), B = (1, 2, - 1)$ and $C = (- 2, - 1, 0)$ find the parametric equations of the plane that they determine.

The vector equation is: $(x, y, z) = (1, - 3, 5) + λ \cdot (0, 5, - 6) + μ \cdot (- 3, 2, - 5)$

Therefore, if we separate component by component we obtain: ${\begin{array}{rcl} x & = & 1 - 3 μ \\ y & = & - 3 + 5 λ + 2 μ \\ z & = & 5 - 6 λ - 5 μ \end{array}$ the parametric equations of the plane.