Generate examples for the intersection of 3 planes Give an example of each case, giving equations of planes in $\mathbb{R}^3$:


*

*Three planes with a common line of intersection

*Intersection by pair, but without common intersection

*Intersection at a single point


Outline of my solution:


*

*System of equations with one free variable (start with a matrix with a line equal to zero and "reverse reduce").

*I have no idea, I only know the system should be inconsistent, but being inconsistent may mean many different configurations.

*System with unique solution (start with identity matrix and "reverse reduce")


Is there a better method to generate the examples than the one I suggest? How could I assure the intersection by pairs in no. 2?
We have not yet studied cross product, only linear systems of equations and dot product. It's the beginning of a linear algebra course.
 A: I am thinking about these problems more geometrically.
Let's start with number 3: Can you think of three very common planes that intersect at a very common point (say, perhaps, the origin?)?
As for one and two, I'd think in two dimensions. Define your planes using only $x$ and $y$ (like for example $x+y=0$). For number 1, choose a point on the $xy$-plane and choose three lines that go through that point. If your point is, say, $(a,b)$, then when you define your equations as planes rather than lines, they will intersect at $(a,b,z)$ for all $z$. Then repeat the process for number 2, but draw a triangle on the $xy$-plane instead of a point.
A: *

*$z=0$, $x=0$, $x+z=0$. The method to find the 3rd one is consider the first two planes you rotate one of them about the y-axis. Then find some points on it geometrically and find the coefficients in the equation $ax+by+cz=d$.

*$z=0$, $x=0$, $x+z=1$ Same method for finding the equation of the 3rd plane here as above.

*$x=0,y=0,z=0$ are equations of 3 planes intersecting at $(0,0,0)$

