We say that a line is parallel to a plane if the direction vector of the line is parallel to the plane. Let be the line containing the point and parallel to the planes

Find a vector parametric equation for .

The normal vectors of the planes are and . So, the direction vector of will be perpendicular to both of these,

From the first equation we have . Plugging this into the second equation we obtain , which then gives us . Since is arbitrary, we take to obtain a direction vector . Therefore, the vector parametric equation for the line is