1. The eight vertices of a cube centered at (0, 0, 0) of side length 2 are at (±1, ±1, ±1).
a) Find the four vertices of the cube, starting with (1, 1, 1), that form a regular tetrahe- dron. Confirm your answer by finding the length of an edge and explaining why all edges have the same length.
b) A methane molecule consists of a hydrogen atom at each of the vertices of a regular
tetrahedron and a carbon atom at the center. Find the ‘bond angle’, that is, the angle made by vectors from the carbon atom to two hydrogen atoms (use a calculator; round your answer).
c) Use the dot product to find the angle between two adjacent edges (sharing a common vertex) of the tetrahedron; and the angle between two opposite edges. Explain your answers using symmetry.
d) Find the area of a face of the tetrahedron.