Linear Algebra and Its Applications · Problem 3.2.8
Orthogonality — Gilbert Strang
Mathematics · cosines-projections-onto-lines
Problem
The methane molecule CH4 is arranged as if the carbon atom were at the center of a regular tetrahedron with four hydrogen atoms at the vertices. If vertices are placed at (0,0,0), (1,1,0), (1,0,1), and (0,1,1)---note that all six edges have length 2, so the tetrahedron is regular---what is the cosine of the angle between the rays going from the center (21,21,21) to the vertices? (The bond angle itself is about 109.5°, an old friend of chemists.)