Consider the equation above that states $\mathbf{P} = [\mathbf{M} | -\mathbf{Mc}]$. If I were to multiply this matrix with a vector $\mathbf{\tilde{c}} = [c_x, c_y, c_z, 1]$, the result is $\mathbf{P}\mathbf{\tilde{c}} = \mathbf{M}\mathbf{c} - \mathbf{M}\mathbf{c} = 0$ where $\mathbf{c} = [c_x, c_y, c_z]$.
Why is it that Pc = 0?
Consider the equation above that states $\mathbf{P} = [\mathbf{M} | -\mathbf{Mc}]$. If I were to multiply this matrix with a vector $\mathbf{\tilde{c}} = [c_x, c_y, c_z, 1]$, the result is $\mathbf{P}\mathbf{\tilde{c}} = \mathbf{M}\mathbf{c} - \mathbf{M}\mathbf{c} = 0$ where $\mathbf{c} = [c_x, c_y, c_z]$.
That makes sense, thanks!