Four Coplanar Points: Solving for Unknown X

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In summary, the person is trying to figure out how to show that four points are coplanar, with one of them having an unknown x-coordinate. They mention using cross products and triple products to determine if the points lie in the same plane. They also consider finding an equation for the plane and using the distance between the points to determine if they are coplanar. Lastly, they suggest finding the volume of the tetrahedron defined by the points, as it will be 0 if they are coplanar.
  • #1
Safy91
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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).
 
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  • #2
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. :(
 
  • #3
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.
 
  • #4
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.
 
  • #5
Safy91 said:
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.
 
  • #6
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.
 

What is a coplanar point?

A coplanar point is a point that lies on the same plane as other given points. In other words, all the points are on the same flat surface.

How do you determine if four points are coplanar?

To determine if four points are coplanar, you can use the coplanar point theorem. This theorem states that if three points are coplanar, then any additional point will also be coplanar with the first three points. So, if you can draw a line connecting each point to one another without crossing any other points, then the points are coplanar.

What is the formula for solving for unknown x in four coplanar points?

The formula for solving for unknown x in four coplanar points is x = (b*d - a*c) / (b - a), where a, b, c, and d represent the x-coordinates of the four points.

Can you use the same formula to solve for unknown y?

No, the formula for solving for unknown x in four coplanar points only applies to the x-coordinate. To solve for unknown y, you would need to use a similar formula but with the y-coordinates of the points.

Why is solving for unknown x important in four coplanar points?

Solving for unknown x in four coplanar points can help determine the position of a point on a plane. This is useful in various fields such as geometry, engineering, and navigation. It allows for precise calculations and measurements to be made.

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