Determine whether a set of points lie on the same plane

  • #1
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Homework Statement:

All below

Relevant Equations:

All below
1594693398811.png

AB, AB, AD are Ld, that is, the three vectors lie on the same plane, so, "yes, the points lie on the same plane"
However,
AB CB and AD are Li, that is, the three vectors span the space R3, and dont lie in the same plane, so, how can four points that lie on the same plane, that can generate only vectors that lie on the plane, generate a space?
 

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  • #2
anuttarasammyak
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Any three points, say A,B and C, are on the same plane, say P. Is the fourth one, D, is also on the plane P?
If the vectors satisfy the conditions you mentioned, yes it is.
 
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  • #3
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Any three points, say A,B and C, are on the same plane, say P. Is the fourth one, D, is also on the plane P?
If the vectors satisfy the conditions you mentioned, yes it is.
AB AD AC are LD, but AB CB and AD not!

But if A B C D are coplanar, AB CB and AD would need to lie in the plane too!

It are contradictory, no?
 
  • #4
vela
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In the solution, there's a sign error in ##\vec b##. Once you fix that, the determinant ends up not vanishing.
 
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