Four Coplanar Points: Solving for Unknown X

[Safy91]

So I have 4 points, one of them has an unknown x co-ordinate. How would I show that they are coplanar? With 3 points I could just turn them into vectors, work out the cross product and conclude they lie in the same plane if the answer is 0, right? But the 4th point and the unknown is throwing me. Any hints?

Edit: Well, I need to eventually work out what value of the unknown x would satisfy, um, co-planarity (you know what I mean).

 

[Safy91]

Would it be possible to show that the 3 definite points are co-planar, and then use the 4th along with 2 others to work out what the unknown would have to be?

Edit: No, that wouldn't work. :(

 

[Safy91]

Ok how about using one point as the origin, and then finding a value for the unknown that would ensure the cross product of the remaining three is 0?

Umpteenth edit: Er, I don't think that would work either.

 

[Safy91]

Ok, maybe I need some sort of equation for the plane of the first three points, and then just need to find a value that shows the 4th point lies on the plane (I assume it would be 0 distance away). I'm not sure how to represent the plane in cartesian form.

 

[Safy91]

Ok how about using one point as the origin, and then finding a value for the unknown that would ensure the cross product of the remaining three is 0?

Umpteenth edit: Er, I don't think that would work either.

Um, that should have been triple product, not cross. I guess this is probably worth a shot.

Talking to yourself can be helpful, it seems.

 

[AlephZero]

Find the volume of the tetrahedron defined by the four points. If they are coplanar the volume is 0.

You can do that either by finding a triple product, or evaluating a 4x4 determinant.