Plane transformations on cartesian cooordinates

Question

I've noticed that if we have a plane $Αx+By+Γz+Δ=0$ that the term $Δ$ works like a shift of the plane.

For example if $(π_1): x+y+z=0$ and $(π_2): x+y+z=2$, then $(π_2)$ is simply the same as the first one but shifted $+2$ on every axis.

My questions are

1. if we have a plane $(p): 2x-y+3z=4$, then to find out by how many units it shifted on the axes, I simply do this? $xx': \frac{4}{2}, yy' : \frac{4}{-1}, zz': \frac{4}{3}$ (which is simply finding the coordinates of the points of intersections with the axes?)

2. are there similar important transformations of planes I should be aware of? Say, what the coefficients do on each unknown?

Answer 1

Gometrically, a change in the independent term draws another plane parallel to the first. To your point 1, you are considering two planes $2x-y+3z=0$ and $p:2x-y+3z=4$. Their normal vectors are the same, i.e. $\vec d=(2,-1,3)$ meaning they are parallel. An interesting property is what you call "shift", but measured as the minimum distance between them. You can use this to calculate it:

$$D=\frac{ax_0+by_0+cz_0+d}{\sqrt{a^2+b^2+c^2}}$$

Being $(x_0,y_0,z_0)$ the coordinates of the point you want to calculate the distance of it to the plane and $a,b,c$ the coefficients. You can choose the point $(x_0,y_0,z_0)=(0,0,0)$ of $x,y,z$ belonging to the first plane to calculate the distance between them (it may be negative in the general case, you can drop the sign if necessary):

$$D=\frac{a·0+b·0+c·0+d}{\sqrt{a^2+b^2+c^2}}=\frac{-4}{\sqrt{2^2+(-1)^2+3^2}}=\frac{-4}{\sqrt{14}}$$

Anyway, you are correct, the new plane have all the points of intersection of the axes shifted by the same amount: the property they exhibit is that they are parallel.

The second question about other transformations, I can think of a rotation and displacement combined. Often is necessary consider perpendicularity between planes, easy to treat because the planes' normal vectors are perpendicular too and their dot product is null. e.g. for your plane $p$, the plane $q:-x+y+z=0$ is perpendicular: $(-1,1,1)·(2,-1,3)=0$