# Linear algebra: For which values of 'a' are P = (-1, 1, 2), Q = (0, a, 1), R = (a, 4, -1) and S = (-11, -1, 0) corners of a tetrahedron?

#### pingaan

##### New member
Exercise:
For which 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 of a tetrahedron? Cacluate the volume of the tetrahedron PQRS for these values of a.

I really need some advice on how to execute this on, as I have no idea where to begin.

Best regards.

#### Jomo

##### Elite Member
What are the properties of a tetrahedron? Are the sides all the same length?
Can you please show us what you tried? That way we know where you are stuck.

#### pingaan

##### New member
The exercise is stated exactly like that. The form could be symmetric, but doesn't have to be.

I have tried simply calculating the volume (V=1/6*(A-B+C) with the values, but ended up with countless of decimals.
I have tried setting up a plane on one of the sides and gauss eliminate - didn't quite work out.
I have tried vector sum on one of the sides in order to find a value of a.

Nothing of the things I've tried has gotten me anywhere.

#### Jomo

##### Elite Member
I would try finding out the length of each side (Hint: they are all equal)

#### pingaan

##### New member
How do you know that they are? I mean, determining that all sides are equal would be the hardest step.

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#### Dr.Peterson

##### Elite Member
I don't think the question is about a regular tetrahedron, since it doesn't say that. I think it's asking about the points not being collinear or coplanar, so that they can be the vertices of a polyhedron. In fact, if you find the volume (you may know how to do that using vectors), the volume being non-zero would then be the condition for the first question.

But I could be wrong. This is the first step in solving the problem: to state your interpretation of what is being asked, and therefore determine what the condition is that you are solving for.

#### pingaan

##### New member
I sorted it out. It was quite simple when knowing how to terminate it; simply calculate the volume and determine the roots of a.

a≠2,−18

#### Dr.Peterson

##### Elite Member
Right. If it has volume 0, then it isn't a tetrahedron.