Is it possible to add/multiply planes?

Question

Is it possible to add or multiply planes?

For example if I have $x+y+z=0$ in $\mathbb{R}^3$, is it possible to do something like the following? $${(x+y+z=0)+(x+y+z=0)}={(2x+2y+2z=0+0)}$$ $${(x+y+z=0)\cdot(x+y+z=0)}={(x^2+y^2+z^2+2(xy+xz+yz)=0\cdot0)}$$

Please excuse my (probably) dreadful notation - I wasn't sure how to represent this concept.

Any feedback is greatly appreciated.

Answer 1

Yes you can do those operations. As you have already noticed, planes can be represented by equations and naturally, you can add or subtract or multiply the equations to each other.

For example, we can use addition/subtraction of two plane equations while finding the intersection of those planes (in this case, the planes are the same but assume planes are different).

Answer 2

In the addition bit, you actually end up getting the same plane you had before: what you did was multiply your equation by two. You can actually "add" planes in a sense: given $\pi_1$, $\pi_2$ two affine planes, $\pi_1+\pi_2$ is defined as the smallest affine variety $Y$ so that $\pi_1,\pi_2\in Y$.

As for the multiplication part, that is perfectly correct in a geometric way: what you got by multiplying the two equations is the equation of a "double plane", which is a (degenerate) conic (a conic such as the ellipse, the parable...). If you are interested in the topic, you can research quite a lot about conics in any book about non-linear geometry, specially books about Projective Geometry.