Linear algebra: Tetrahedral

[Oleg]

Hi, so I have a problem which goes:

For what values of a∈R are the points P=(-1,1,2) Q=(0,a,1) R=(a,4,-1) and S=(-11,-1,0) corners in a tetrahedral? Determine the volume of the tetrahedral PQRS for these values of a (positive oriented orthonormal system).

So I was thinking that I needed to make all the lengths PQ, PR, PS, QR, QS and RS equal since it is a tetrahedral. Since PS is the only one without the unknown a, I solve it to determine how long all the other sides should be and I get PS=sqrt((-11-(-1))^2+(-1-1)^2+(0-2)^2) = sqrt(108). I then find expressions for the remaining sides and put them equal to sqrt(108). But here I run into a trouble, between the different sides, I don't get any value for a which is true for all of them.

For example PQ=sqrt((-1)^2+(a-1)^2+(-1)^2) = sqrt(a^2-2a+3) and PR=sqrt((a+1)^2+3^2+3^2) = sqrt(a^2+2a+19)

This gives sqrt(108) = sqrt(a^2-2a+3)
108 = a^2-2a+3
a = 11.295630140987 or a = -9.295630140987

and from sqrt(108) = sqrt(a^2+2a+19)
108 = a^2+2a+19
a = 8.4868329805051 or a = -10.486832980505

So I am not sure what to do, since I can't seem to find any valid value for a.

Also, I'm not sure if I understand the part about finding the volume of the tetrahedral depending on a. Isn't the length of the sides (sqrt(108)) the only thing that matters in terms of volume and it shouldn't matter where a happens to be positions?

 

[tkhunny]

1) A tetrahedron has a tetrahedral shape. Difference between a noun and an adjective. (Unless there is some cultural difference I am missing.)

2) How shall you demonstrate whether it is or is not a tetrahedron? Recall from elementary geometry that three points do NOT define a triangle if the three points are collinear. Can we use an analog of that to talk about proving it is a tetrahedron?

3) Why must a tetrahedron have all edges of equal length? The problem statement doesn't specify a REGULAR Tetrahedron, does it?

4) Please find vector methods to calculate the volume.

 

[Dr.Peterson]

A tetrahedron is not necessarily a regular tetrahedron, though that is a common mistake.

I think what you need is to make sure the points are not coplanar or collinear, as those are what would make it not a tetrahedron.

Do you know the formula for the volume of a tetrahedron given the coordinates of its vertices? If you don't know it in that form, you may know it in terms of vectors.

@tkhunny: Great minds ...