Can Any 2-Plane Be Mapped to Another Using Linear Maps and Translation?

[WWGD]

Hi, just curious as to whether we can map any 2-planep: ax+by+cz=d into any other

2-plane p': a'x+b'y+c'z=d' by using a linear map (plus a translation , maybe). I was thinking

that we could maybe first translate to the origin , for each plane, then , given the

angles ( t,r,s) with the respective x,y,z axes, we could rotate by (-t,-r,-s) to have

a plane z=constant , and do the same for c'. Would that work?

 

[micromass]

First translate each plane to the origin. Then take a basis for each plane and take a linear map which maps the basis vectors to each other.

 

[WWGD]

But how do you find a basis for a plane using only the equation ax+by+cz=0?

 

[micromass]

But how do you find a basis for a plane using only the equation ax+by+cz=0?

Well

$$(-b/a,1,0),(-c/a,0,1)$$

is a basis (if a is nonzero). If a is zero, then you must do something analogously with b and c.