Finding Molecule angles with vectors

[relativitydude]

I posted this in the chemistry section, but I was recommended up here. We have a molecule with three unbonded electon pairs, so they will repel each other as much as possible. That's easy, it's 360º/3 = 120º

However, when we have four unbonded electron pairs the angle is 109.5º How?

I figured there are four vectors:

Vector A: (X1)i + (y1)j + (z1)k
Vector B: (X2)i + (y2)j + (z2)k
Vector C: (X3)i + (y3)j + (z3)k
Vector D: (X4)i + (y4)j + (z4)k

And if they repelled each other, it would simply be A + B + C + D = 0

I am figuring setting up this further by dotting the vectors and dividing by the norm to get an algebraic quantity representing the angle. The sum of the angles around XY would be 360º, XZ would be 360º, YZ would be 360º

Then squaring those individual XY, XZ, and YZ and taking the root of it to get 109.5º

 

[relativitydude]

Dare no one help?

 

[mu]

The electron pairs will form a tetrahedron. Knowing that, you can find the angles using simple trigonometry. I'm in a bit of a hurry right now, but I can post details later if you want.

 

[lalbatros]

see the calculations there:

http://mathworld.wolfram.com/Tetrahedron.html

the angle you look for is 90° + (angle of formula 15)