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

  1. Three planes with a common line of intersection
  2. Intersection by pair, but without common intersection
  3. Intersection at a single point

Outline of my solution:

  1. System of equations with one free variable (start with a matrix with a line equal to zero and "reverse reduce").
  2. I have no idea, I only know the system should be inconsistent, but being inconsistent may mean many different configurations.
  3. 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.

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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.

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  1. $z=0$, $x=0$, $x+z=0$. 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 $ax+by+cz=d$.
  2. $z=0$, $x=0$, $x+z=1$ Same method for finding the equation of the 3rd plane here as above.
  3. $x=0,y=0,z=0$ are equations of 3 planes intersecting at $(0,0,0)$
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