a,b,c are the position vectors of A,B,C respectively.Prove that( a x b)+(b x c)+(c x a) is a vector perpendicular to plane of triangle ABC.

Let p = ( a x b)+(b x c)+(c x a) ,

Now , p is perpendicular to the plane only if AB.p = BC.p = CA.p = 0

 

Now AB.p = (b-a).p , On Solving we get , [b c a] - [b c a] = 0

Similarly we can prove it for other vectors too , hence proved.