3D System of Equations No Solution when 3 Planes are Parallel and also Visualizing the Intersection

Question

I am attempting to visualize the intersection between 3 planes. Intersection

I apologize for the bad drawing! In the image above, there are three planes, and the green lines represent the intersection. My confusion is that when a plane intersects with another plane, I don't think the intersection line slopes up or down, so it never intersects with the other intersection lines (i.e. the green lines in the image above). I think I'm incorrect but I'm struggling to prove why I'm wrong.

Also, I read that if three planes are "perpendicular to the page", the intersection forms a "triangle" shape and therefore there's no intersection. I can't visualize this.

Any help to my above questions would be strongly appreciated. Please let me know if I need to further clarify anything.

Answer 1

First, while the "planes" are so poorly drawn in your picture that it is hard to make sense of it, the green lines do not appear to have any relationship to the intersections of those planes. If you draw the lines that best represent the intersections of those planes, as I did in blue, they appear to be very close to intersection, with the difference easily attributable to the roughness of the drawing:

More generally, to expand a bit more on the comments:

For any two planes, there are two possibilities for their intersection:

• The two planes are parallel, and so there is no intersection.
• The two planes are not parallel, and intersect in a line.

Suppose we have three distinct planes $A, B, C$, and that neither $B$ nor $C$ is parallel to $A$. That means both $A\cap B$ and $A \cap C$ are lines. In fact, they are lines in the plane $A$. At this point, the situation is much easier to visualize. We can draw lines in the plane $A$.

There are three cases:

• $A\cap B$ and $A\cap C$ are the same line. For example, in coordinate space, the planes $x = 0, y = 0,$ and $y = x$ intersect in the $z$-axis. Clearly this common line of intersection is in $A \cap B\cap C$. If there were any other point in common to all three planes, then all the planes would be the same, contrary to the assumption that they were distinct.
• $A\cap B$ and $A\cap C$ are separate, but parallel, like the lines $u$ and $v$. This will happen if $B$ is parallel to $C$. It can also happen if $B\cap C$ happens to be parallel to $A$. This is the case in your "all three planes are perpendicular to the page" example. In all cases, when $A\cap B$ and $A\cap C$ are parallel, $A\cap B\cap C$ is empty, as any point in it would lie on both
• $A\cap B$ and $A\cap C$ are not parallel, intersecting like the lines $u$ and $w$. In this case, $A\cap B\cap C$ is the single point of intersection. An obvious example are the three coordinate planes $x = 0, y = 0, z = 0$, whose intersection consists of only the origin $(0,0,0)$.

That exhausts the possibilities for $A, B, C$ having no parallel planes, or at most one pair of parallel planes. If there are two pairs of parallel planes, then all three planes must be parallel to each other, and thus they cannot intersect.