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\$:

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