MathLabs

Problem 6

Given four distinct parallel planes, prove that there exists a regular tetrahedron with a vertex on each plane.
Step 4 of 5: Scale and orient the tetrahedron
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u=n∥n∥,s=∥n∥u=\dfrac{n}{\lVert n\rVert},\quad s=\lVert n\rVert
Detailed analysis

Let u be the unit vector in the direction of n and scale the reference tetrahedron by s. Orient it so that u is the unit normal to the parallel planes, and put its first vertex on the reference plane.