Bond angle in methane? Have been doing for 1 hr!!

[elpermic]

1. The problem statement, all variables and given/known data
In the methane molecule, CH4, each hydrogen atom is at a corner of a regular tetrahedron with the carbon atom at the center. In coordinates where one of the C-H bonds is in the direction of i + j + k, an adjacent C-H bond is in the i - j - k direction. Calculate the angle between these two bonds?



2. Relevant equations
I know this is a pyramid.



3. The attempt at a solution

[rl.bhat]

If A and B are vectors, then the angle between them is given by
cosθ= A.B/A*B

[elpermic]

Would that mean that cos theta is equal to the (Ax+Bx) + (Ay+By) divided by the magnitudes of A and B? I think that each side of the pyramid is 1. Is this right?

[rl.bhat]

Would that mean that cos theta is equal to the (Ax+Bx) + (Ay+By) divided by the magnitudes of A and B? I think that each side of the pyramid is 1. Is this right?
No.
cosθ = (AxBx + AyBy + AzBz)/AB, where A and B are the magnitudes of the vectors.

[elpermic]

So how would I use the method of components to find Ax and Bx, and Ay and By?? I never learned how to do Az and Bz either. I only have one fact that I think, each side is 1.

[rl.bhat]

You have expressed bonds is in the direction of i + j + k, an adjacent C-H bond is in the i - j - k direction.
Here coefficient of i, j and k are Ax, Ay and Az. i.e. 1, 1, 1.
Similarly find Bx, By and Bz. Magnitudes of A and B are sqrt(3)
Now find the angle using the formula.

[Phrak]

It might help to know that a tetrahedron can be embedded in a cube. I think it's far more intuitive to obtain the H-C-H angle by examining the cube.

http://math.ucr.edu/home/baez/dodecahedron/anim5in1.gif

umm. that wasn't supposed to be spinning, but it's the best I could find.

[elpermic]

I have been trying it out and I still cannot do this

[Phrak]

Consider if the cube is one unit on a side. The blue line at the top of the cube connects two hydrogen atoms. The length of the line is sqrt(2).

The center of this line lies at the center of a cube face. The center of the cube face is 1/2 unit from the center of the cube. The carbon atom is at the center of the cube.

Draw a picture. Does the line from the center of the cube intersect the blue line at a right angle?