Mathematics 41, section 1 -- Multivariable Calculus

Discussion 2 -- A Regular Tetrahedron

September 10, 1999

Background

A regular tetrahedron is a 3D polyhedron with four planar equilateral triangular faces meeting along six edges (all of the same length), with four vertices, or corner points, where three faces come together. Tetrahedra occur in nature, for example, in the the methane (CH₄) molecule. The four hydrogen atoms in a methane molecule are (approximately) the vertices of a regular tetrahedron, with the carbon atom at its center. Today we will use the geometry of vectors to study one example of a regular tetrahedron and to determine the bond angle between any two of the carbon-hydrogen bonds in a methane molecule.

Discussion Questions

All the following questions refer to the tetrahedron in R³ with vertices at the points P = (0,0,0), Q = (1,1,0), R = (1,0,1), and S = (0,1,1):

• A) The six edges of the tetrahedron are the line segments PQ, PR, PS, QR, QS, RS. Show that all of these edges have the same length.
• B) Give a vector description of the edge of the tetrahedron containing the points Q, S
• C) Find the equations of the plane containing the points Q,R,S by first finding a normal vector to that plane, then using our general form for the equation of a plane with given normal, containing a given point. Does it matter which of the points Q, R, S you use as the given point?
• D) Verify from your answers to parts C and B that the edge QS lies in the plane containing the triangular face with corners Q,R,S . (How?)
• E) The center of this tetrahedron is at the point C = (1/2,1/2,1/2) . What is the angle between the vectors v from C to P and w from C to Q. Express your result in both radians and degrees. Also, explain why this angle should be the same as the angle between any two carbon-hydrogen bonds in a methane molecule.

Assignment

Group write-ups due in class, Wednesday, September 15.