Question

Introduction: Throughout the course the assignments tie important concepts to applications, sometimes mathematical but sometime veering (not too deeply) into other fields. This week we consider a chemistry application. Methane is a chemical compound with the symbol CH4, having four hydrogen atoms and one carbon atom. The molecule is modeled in a tetrahedral shape, with the four hydrogen atoms connected to a central carbon atom (by C-H bonds, if you know a little chemistry). The goal is to determine the angle between the bonds. The answer is, of course, well known–you may know it already, if you are a chemist! If not feel free to look it up to check your answer, but you should write up a clear solution.

You should not use any coordinates in your solution! View the bonds as vectors v1, v2, v3, and v4 from the carbon atom at the origin O to the four hydrogen atoms.

1. What assumption(s) is (are) reasonable for the magnitude of the vectors? Explain.

2. What can we say about the sum of the four vectors? Briefly support your answer and give an equation involving the vi .

3. Mathematically manipulate your equation to obtain a scalar equation instead of a vector equation. (Hint: what operation can turn a vector into a scalar?)

4. Solve your scalar equation to find the angle. Show all appropriate work.

Answer 1

Calculate the angle between two verticesof the regular tetrahedron, lets pick any two of them and draw vectors to them from the center. As you can see from above fig. I chose points (0,1,1) and (1,0,0). We just have to find the angle between the two constructed vectors.So,

Using the dot product to compute the angle between and ,