Video: Intersection between Planes

What is the intersection of the plane through 𝐴𝐡𝐡′𝐴′ and the plane through 𝐡𝐢𝐢′𝐡′?

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Video Transcript

What is the intersection of the plane through 𝐴𝐡𝐡 dash 𝐴 dash and the plane through 𝐡𝐢𝐢 dash 𝐡 dash?

Let’s first consider the plane 𝐴𝐡𝐡 dash 𝐴 dash as shown in the diagram. Well this plane has four edges 𝐴𝐡, 𝐡𝐡 dash, 𝐡 dash 𝐴 dash, and 𝐴 dash 𝐴 and four vertices: 𝐴, 𝐡, 𝐡 dash, and 𝐴 dash.

Now let’s consider the second plane, 𝐡𝐢𝐢 dash 𝐡 dash. This plane also has four edges 𝐡𝐢, 𝐢𝐢 dash, 𝐢 dash 𝐡 dash, and 𝐡 dash 𝐡. It also has four vertices: 𝐡, 𝐢, 𝐢 dash, and 𝐡 dash.

Where do these two planes intersect? Do they intersect at a vertex or point or do they intersect at an edge? Well it is clear from the diagram that the two planes intersect at 𝐡𝐡 dash.

Therefore, the intersection of the plane through 𝐴𝐡𝐡 dash 𝐴 dash and the plane through 𝐡𝐢𝐢 dash 𝐡 dash is the line or edge 𝐡𝐡 dash. You will notice that when looking at the original two planes, the two vertices that appear in both of them are 𝐡 and 𝐡 dash.

This means that the planes must intersect at 𝐡 and 𝐡 dash. We could write this as the intersection of 𝐴𝐡𝐡 dash 𝐴 dash and 𝐡𝐢𝐢 dash 𝐡 dash is 𝐡𝐡 dash.

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