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.

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