Finding the inside angle of a tetrahedron

[xaer04]

Homework Statement

"ASSIGNMENT 1
The Methane Molecule

Introduction: The methane molecule CH4, composed of four hydrogen atoms and one carbon atom, is shaped liked a regular tetrahedron. The four hydrogen atoms are on the vertices and the carbon atom is at the center. What is the angle between the CH bonds?

The fact that all the hydrogen atoms will behave the same allows us to make several simplifying assumptions. For example, all four of the inside angles HCH will be the same, so we only need to calculate one of them.

Your Task: Demonstrate how the angle can be found using both of the following methods. Give your answers in degrees and make sure it accurate to one place past the decimal. (If you get two different answers there is a good chance one of them is wrong!)
I. Find the coordinates of the regular tetrahedron, then use your knowledge of vectors to calculate the angle. Use a constructive method and show how you calculate the coordinates. For this part you may assume all the vectors have length one.
II. To get started for the second method, assume the coordinate system is chosen so that the carbon atom is at the origin. The four CH bonds can be viewed as four vectors. Notice the following facts:
a. All the vectors have the same length, since the bonds are identical.
b. The sum of all four vectors is zero. If this were not the case then there would be a net force and the atoms would be unstable.

To calculate the angle, start by writing an equation involving the four vectors. Next, in a fit of inspiration take the dot product of (both sides of) the equation with the one vector. Applying what you know about dot products, you will get an equation involving the angle . For this part you should complete the whole calculations without using the vector components for any step."

$$\vec{A} + \vec{B} + \vec{C} + \vec{D} = 0$$

Homework Equations

$$\vec{a}\cdot\vec{b} = \mid\vec{a}\mid\mid\vec{b}\mid \cos{\theta}$$

The Attempt at a Solution

I looked up the answer and it's supposed to be $109.5^{\circ}$, but i have no idea where to begin.

The picture is from wikipedia... I'm not too great at drawing in 3d.

 

[Dick]

Take the dot product of e.g. A with the vector equation. And start thinking about what the relations are between the parts.