Equation of Plane Passing through the Intersection of Two Planes

If two planes $\vec r\cdot\vec n_1 - d_1 = 0$ and $\vec r\cdot\vec n_2 - d_2 = 0$ actually intersect (i.e. their normals are not parallel), then for every real value of $\lambda$ the equation $\vec r\cdot(\vec n_1 + \lambda\vec n_2) = d_1 + \lambda d_2$ represents some plane passing through their common line of intersection. As $\lambda$ runs over all real numbers, this single formula sweeps out the whole pencil (family) of planes hinged along that line — Fig. 6.12 pictures the two given planes as two sheets meeting along a shared line, with every other plane through that line being one more sheet through the same hinge.

The same idea in Cartesian coordinates: if $a_1x+b_1y+c_1z+d_1=0$ and $a_2x+b_2y+c_2z+d_2=0$ intersect, then for every real $\lambda$, $(a_1x+b_1y+c_1z+d_1) + \lambda(a_2x+b_2y+c_2z+d_2) = 0$ represents a plane through their line of intersection.

This is useful precisely because it turns "find the plane through this line and satisfying one more condition" into a one-parameter search: write down the family, plug in the extra condition (typically a third point the plane must pass through), solve for the single unknown $\lambda$, and substitute back. …