The plane Π contains the points (1, 2, 3), (0, 1, 2) and (2, 3, 0). What is the vector equation of the plane? and what is the cartesian equation of the plane?

Vector Equation So we know it contains three points so we can find two lines in the plane. 1) (1,2,3) + A((0,1,2) - (1,2,3)) = (1,2,3) + A(-1,-1,-1) 2) (1,2,3) + B((2,3,0) - (1,2,3)) = (1,2,3) + B(1,1,-3) Generally the vector form of a plane will be in the form of a point on the plane and two different direction vectors, so we can deduce from above that one possible plane equation with these 3 points is (1,2,3) + A(-1,-1,-1) + B(1,1,-3) Cartesian Equation So first lets find the normal to the plane. We do this by doing the cross product of the two direction vectors in the vector equation (-1,-1,-1)X(1,1,-3) = (4,-4,0) If we take out a factor of four due to it being a direction vector we end up wih the normal being (1,-1,0) so the general form of the cartesian is (a,b,c)^.(x,y,z)=d or ax+by+cz=d where (a,b,c) is the normal to the plane and d is the product of a point on the plane replacing (x,y,z) So we can write x - y =d putting in the poiny (1,2,3) gives the full cartesian equation x - y = -1