Steps to creating 3 plane equations with 3 lines of intersection

Question

I was wondering if anyone can give me pointers on to how to mathematically create 3 plane equations that meet in 3 lines. In other words, each plane intersects one another in a straight line (so it sort of makes a triangle in a way)?

Any help would be appreciated!

Answer 1

Hint: Suppose you take the case where all three lines point up, and then consider the normal vectors for the three planes. What do you notice? How would you represent this condition?

Answer 2

I think I may have figured it out (somewhat) but I might still have a few loose ends:

Step 1: Create an equation of a line. This will be used as reference.

Step 2: Find the direction vector of the line.

Step 3: Using the normal of the line and another vector, a, (which you must create yourself), ensure that the dot product equals to zero. This will, thus, mean that you've found the normal of a plane.

Step 4: Find a point on the line.

Step 5: Using the point of the line and vector a (normal of plane 1), you can create your first plane equation

Step 6: Since the normal of one plane is a linear combination of the other two, find two other noramls and create another 2 plane equations

I just don't think it's possible to use the same point on the line so I'm stuck.