Video: Finding the Intersection between Three Given Planes

Determine the intersection of the plane 𝐷𝐢𝐢′𝐷′ and plane 𝐴𝐡𝐢𝐷 and plane 𝐴𝐷𝐷′𝐴′.

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Video Transcript

Determine the intersection of the plane 𝐷𝐢𝐢 dash 𝐷 dash and plane 𝐴𝐡𝐢𝐷 and plane 𝐴𝐷𝐷 dash 𝐴 dash.

In this question, we’re looking for the vertex or edge at which the three planes intersect. Let’s first consider the plane 𝐷𝐢𝐢 dash 𝐷 dash and the plane 𝐴𝐡𝐢𝐷. Well, these two planes intersect along the edge 𝐢𝐷. This is because they have a common edge 𝐢𝐷 belongs to the plane 𝐷𝐢𝐢 dash and 𝐷 dash and also belongs to the plane 𝐴𝐡𝐢𝐷.

However, we also need to consider the plane 𝐴𝐷𝐷 dash 𝐴 dash. We can see from the diagram that the plane 𝐴𝐷𝐷 dash 𝐴 dash intersects 𝐴𝐡𝐢𝐷 along the edge 𝐴𝐷. The plane 𝐴𝐷𝐷 dash 𝐴 dash also intersects the plane 𝐷𝐢𝐢 dash 𝐷 dash along the edge 𝐷𝐷 dash.

We were asked to determine the intersection of all three planes 𝐷𝐢𝐢 dash 𝐷 dash intersects 𝐴𝐡𝐢𝐷 intersects 𝐴𝐷𝐷 dash and 𝐴 dash at the point 𝐷. This is because 𝐷 is the only common vertex of all three planes. 𝐷 exists in the plane 𝐷𝐢𝐢 dash 𝐷 dash. It also exists in the plane 𝐴𝐡𝐢𝐷. And finally, 𝐷 is also a vertex of the plane 𝐴𝐷𝐷 dash 𝐴 dash.

This means that the point of intersection is the vertex 𝐷.

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