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 tetrahedron.Confirm your answer by finding the length of an edge and explaining why all edges havethe same length.(Recall: a tetrahedron is a solid with four triangular faces, like a pyramid with a triangularbase; it is regular if all faces are equilateral triangles. Draw pictures and look at cubicalobjects in order to figure out how equilateral triangles fit on a cube).

b) Use dot product to find the angle between two adjacent edges (edges sharing a common vertex) of the regular tetrahedron; and the angle between two opposite edges (edges that lie on skew lines; even though they dont intersect, you can still compute the angle made by their directions). Explain your answers using symmetry.

c) A methane molecule CH4 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, i.e. the angle made by vectors from the carbon atom to two hydrogen atoms.