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?

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pingaan

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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.
 
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.
 
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.
 
I would try finding out the length of each side (Hint: they are all equal)
 
How do you know that they are? I mean, determining that all sides are equal would be the hardest step.
 
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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.
 
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
 
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