Affine coordinates of a line

Question

Can you help me figuring out how to solve the next problem?

If the points M and N have affine coordinates $(m_1,m_2,m_3)$ and $(n_1,n_2,n_3)$ with respect to some points A,B,C, then the points X of the line MN have the affine coordinates $(x_1,x_2,x_3) = m (m_1,m_2,m_3) + n (n_1,n_2,n_3)$ with respect to A,B,C. Prove that, up to a scalar multiple, there exists an unique triplet $(p_1,p_2,p_3)$ of real numbers having the property: $(p_1,p_2,p_3) * (x_1,x_2,x_3) = 0$ for every X belonging to MN.

Thank you.

Answer 1

The legal sets of affine coordinates form the affine plane $P=\{\,(x,y,z)\mid x+y+z=1\,\}\subset\Bbb R^3$, and the line $MN$ is a line in$~P$. The vector subspace$~V$ generated by the elements of $MN$ is the unique plane through the origin containing $MN$. For any such plane there exist $(p_1,p_2,p_3)\neq(0,0,0)$ such that $V=\{\,(x,y,z)\mid p_1x+p_2y+p_3z=0\,\}$, and these are unique up to multiplication by a common nonzero scalar (this is a standard representation for vector hyperplanes). The same equation $p_1x+p_2y+p_3z=0$ characterises $MN=V\cap P$ inside $P$.