Test for coplanarity of four points

[Nero26]

Hi all,
If a,b,c,d are position vectors of four points A,B,C,D.The points will be coplanar if xa+yb+zc+td=0,x+y+z+t=0,provided x,y,z,t are not all 0,and they are scalars.Is this test needed to show 4 points are coplanar?
If we consider two lines joining A,B and C,D then this will give us two vectors which are always coplanar.So points A,B,C,D are also coplanar.So I assumed that any 4 points are coplanar and no test is needed for it.
Or is this the test to verify coplanarity of D with the plane containing A,B,C ?
I'm wondering if my assumption is true?Please help me clarifying it.
I'm new here ,Please treat my mistakes with forgiveness.
Thanks.

 

[LCKurtz]

Hi all,
If a,b,c,d are position vectors of four points A,B,C,D.The points will be coplanar if xa+yb+zc+td=0,x+y+z+t=0,provided x,y,z,t are not all 0,and they are scalars.Is this test needed to show 4 points are coplanar?
If we consider two lines joining A,B and C,D then this will give us two vectors which are always coplanar.So points A,B,C,D are also coplanar.So I assumed that any 4 points are coplanar and no test is needed for it.

What if A,B,C,D are the vertices of a regular tetrahedron?

 

[Nero26]

What if A,B,C,D are the vertices of a regular tetrahedron?

Thanks a lot for your clue.I think I'm getting near to it.Can you please take a look on the attachment...
And please mention if I need some more things to do.

 

[LCKurtz]

Thanks a lot for your clue.I think I'm getting near to it.Can you please take a look on the attachment...
And please mention if I need some more things to do.

My point was that your statement that any 4 points are coplanar is false. Remember that it takes three non-collinear points to determine a plane (their triangle is part of the plane). Four points are coplanar only if the 4th point lies in the plane determined by the first three.

The test I would use for co-plane-ness of points A,B,C,D would be to make vectors of the sides like this: u = AB, v = AC, w = AD and calculate the triple scalar product or "box" product $u\cdot v \times w$. If that is non-zero they aren't coplanar and if it is zero they are.

 

[Nero26]

Thanks a lot for your help.I think I got your point " Four points are coplanar only if the 4th point lies in the plane determined by the first three."
Have a nice day!