Suppose you have a $3$-dimensional space in which there are 2 points ($A$ and $B$) defined (non identical). Now, you can define a line that goes through them but you cannot define a unique plane, because there are infinitely many planes that are rotating along that line.
Now, I've been having trouble defining the following in strict mathematical terms. That's also why I'm asking here, so you'll have to forgive me for being vague or imprecise.
Suppose that you also chose one axis towards which you want the plane to lay 'flat'. Imagine you model this using table and a piece of paper. The axis along the table are $X$ and $Y$ and the axis up is $Z$. You can rotate the paper along two imaginary points, but at one point it would lay flattest towards your $Z$ axis. There would be a clean slope from one point to the other.
How would you figure out the equation of this plane which would be defined using only two points and an 'flatness axis' (for my lack of better words). My thoughts led me towards using a third point ($C$) which would form a vector ($AC$) perpendicular to the vector ($AB$) where the $Z$ coordinate of point $C$ would be the same as point's $A$. Maybe.
And how would you generalize it to $n$-dimensional cases? (2 points, one flatness axis)
I have managed to reduce my problem to the following question: I have a vector $\vec n$ and $\vec m$. I want to find vector $\vec o$, which is perpendicular to the vector $\vec n$ and lies on the plane defined using vectors $\vec n$ and $\vec m$.