A(a→),B(b→) and C(c→)  are the vertices of triangle ABC and R(r→)  i, any point in the plane of triangle ABC, the r→⋅(a→×b→+b→×c→+c→×a→) i,is always equal to

A vector perpendicular to the plane of  A(a→),B(b→) and C(c→)(b→−a→)×(c→−a→)=a→×b→+b→×c→+c→×a→Now for any point  R(r→) in the plane of A, B and C is (r→−a→)⋅(a→×b→+b→×c→+c→×a→)=0r→⋅(a→×b→+b→×c→+c→×a→) −a→⋅(a→×b→+b→×c→+c→×a→)=0r→⋅(a→×b→+b→×c→+c→×a→)=0→+a→⋅b→×c→+0→r→⋅(a→×b→+b→×c→+c→×a→)=[a→b→c→]