Show that the plane that passes through the three points $A(a_1,a_2,a_3),B=(b_1,b_2,b_3),$ and $C=(c_1,c_2,c_3)$ consists of the points $P=(x,y,z)$ 
Show that the plane that passes through the three points $A(a_1,a_2,a_3),B=(b_1,b_2,b_3),$ and $C=(c_1,c_2,c_3)$ consists of the points $P=(x,y,z)$

Point P can

given by  
So I am stuck on this problem and I think I should use determinants but I'm not sure what that would prove.

First, expand the given determinant by multilinearity on columns. You get 8 determinants, 4 of which are identically 0. The remaining 4 determinants, when you take out the factor of x,y, or z from 3 of them, leave an equation of the form Ax+By+Cz+D=0. So the given determinantal expression is that of a plane. Second, substitute the coordinates of any of the given points into the given determinantal expression, giving a determinant with a row of 0's and therefore equal to 0. So the plane passes through each of the given points. The equation is unique, up to multiplication by a non-zero constant, provided the points are in "general position.'
