I check out all write-ups online concerning how to show four points room coplanar. However, none of them talk about the idea behind the method. Can someone define how the triple scalar product works?




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You know that 3 points $A,B,C$ (two vectors $vecAB$, $vecAC$) form a plane. If you want to present the 4th one $D$ is on the same plane, you have to display that that forms, with any kind of of the other allude already belonging to the plane, a vector belonging to the airplane (for instance $vecAD$).

Since the cross product of two vectors is typical to the plane formed through the two vectors ($vecAB imes vecAC$ is common to the aircraft $ABC$), girlfriend just have to prove her last vector $vecAD$ is typical to this overcome product, for this reason the triple product that should be same to $0$:

$vecAD cdot(vecAB imes vecAC)=0$


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