If $ax + by + cz + d = 0$ is Plane $1$ and $a'x + b'y + c'z + d' = 0$ is Plane $2$, then what does (Plane $1$) + $\lambda$(Plane $2$) signify?
I got this doubt when the equation of a line was given as an intersection of two planes Plane $1$ and Plane $2$. And the general equation of a plane passing through that line was described to be of the form "(Plane $1$) + $\lambda$(Plane $2$)".
How do we arrive at this conclusion? My problem is that I am not able to imagine $\lambda$ times a plane. To my mind it seems to be the same plane itself. But when added with another plane it seems like it might be the equation of all planes passing through the intersection of the two planes. But I am not able to understand why?
Say the plane you are looking for (which passes through the line of intersection (say L) of $P_1$ and $P_2$) be $P_3$. Now, you want that every point on L satisfies the equation of $P_3$.
As equation of $P_3$ is $P1+\lambda P2=0$ and every point on L already satisfies $P_1=0$ and $P_2=0$, it also satisfies $P_3$. Also, you know that $P1+\lambda P2=0$ is a linear equation in variables hence, you know it's a plane. And we have shown that it contains all points on L.
Hence, we are done.