Vector algebra of tetrahedron

If the

mathy:

The vertex length of a tetrahedral geometry from the centre is 'a'. By constructing vecotrs show that resultant of any three vectors is equal and opposite to the fourth vector.

Hi Mathy,
Welcome to web2.0calc forum.

This is totally unfamiliar territory to me so my answer is not likely to be what you really need, but I will talk about it anyway.
I have assumed that this is a regular tetrahedron which is probably incorrect right from the outset. I am sure that i have oversimplified this problem.
Maybe some of the others will be interested in what I say.

I found this site and I have used it to create this 'answer'
http://www.ctralie.com/Teaching/Tetrahedron/

Firstly a tetrahedron is a triangular pyramid.
the site above shows how you can construct a regular one. But I've taken a couple of snips to show you directly.
The regular tetrahedron is drawn inside a cube of side length 1 unit.
Loking at the picture and using pythagoras it can be seen that the side length of this tetrahedron is sqrt2 units.
140105 reg tetrahedron side is root2.JPG

now this is another view of it where you can see that all the vertices are equidistant from (0.5, 0.5, 0.5)
140105 tetrahedron2.JPG

I am going to translate (slide) all the vertices so that they are all equidistant from (0,0,0)
I can do this very simply by subtracting 0.5 off all the x,y and z values.
so
(0, 1, 1) translates to (-0.5, 0.5, 0.5)
(1, 1, 0) translates to (0.5, 0.5, -0.5)
(1, 0, 1) translates to (0.5, -0.5, 0.5)
(0, 0, 0) translates to (-0.5, -0.5, -0.5)

the distance from the 'centre' of each of these points is sqrt(0.5 ²+0.5 ²+0.5 ²) = ... = root3 / 2
I want the distance to be 'a' and I can do that easily enough but it will just mean the vectors are all multiplied by the same constant and when they are added to show what the question asks for, the constant will not impact on the answer at all.
I'll do it anyway.
a = root3/2 x some constant
the constant must be 2a/root3
so the vectors (from the centre) are
1) 2a/root3 (-0.5, 0.5, 0.5)
2) 2a/root3 (0.5, 0.5, -0.5)
3) 2a/root3 (0.5, -0.5, 0.5)
4) 2a/root3 (-0.5, -0.5, -0.5)

Now if you add any 3 of these together you will get the negative of the 4th one.
For example I'll add up the first three and show that the resultant is equal and opposite the forth vector.
2a/root3 (-0.5, 0.5, 0.5) + 2a/root3 (0.5, 0.5, -0.5) + 2a/root3 (0.5, -0.5, 0.5)
= 2a/root3 (0.5, 0.5, 0.5)
= - 2a/root3 (-0.5, -0.5, -0.5)

And that's about all that I have to offer.
I've seen this question posted on another forum, I don't know if it is you who did the other post but you might have more luck over there. (There's a good chance that you will)