If we separate the vector equation component by component we obtain $$$\left\{\begin{array}{rcl} x&=& a_1+\lambda \cdot v_1+\mu \cdot w_1 \\ y&=& a_2+\lambda \cdot v_2 +\mu \cdot w_2\\ z&=& a_3+\lambda \cdot v_3+\mu \cdot w_3\end{array}\right.$$$

which is precisely the parametric equations of the plane.

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: $$(x, y, z) = (1,-3, 5) + \lambda \cdot (0, 5,-6) + \mu \cdot (-3, 2,-5)$$

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