How can i know when points are coplanar?

Question

i already know that if they r in the same plane they r coplanar. but what if 2 points r not in the plane or not in the same plane? How can i tell if they r coplanar? My teacher said if u can draw plane around them... but to me, it sometimes doesn't work.

Answer 1

There isn't much to go on here. Three points determine a plane so if you have such a plane and are given a 4th point and want to know if it lies in the plane, one way to do that would be to calculate the volume of a solid formed by these points. If the volume is 0 then the point is in the plane.

If A, B, C and D are the points. D is the point to determine if it lies in the plane.

AB = vector from A to B
AC = vector from A to C
AD = vector from A to D

The volume is: |AD . (AB x AC)| where "." is the dot product and "x" is the cross product. The cross product gives you a vector perpendicular to the plane and whose magnitude is the area in the plane of a parallelogram formed using AB and AC. When you take the dot product using AD this is then the height of a solid times the area of it's base (dot product is the magnitude of each vector times the cosine of the angle between them).

If |AB . (AC x AD)| = 0 then D lies in the plane since there is no sold.

Is this the sort of answer you are looking for?