The points {3,3,0}, {0,6,3} and {6,6,7} all lie on the same plane. Find the Cartesian equation of the plane.

A Cartesian equation is in the form ax + by + cz = d. The a, b and c constants are given by the normal vector to the plane so we must find this first. You find the two direction vectors of the plane by subtracting any two of the vectors from the other one, eg: [3,3,0] - [0,6,3] = [3,-3,-3] and [3,3,0] - [6,6,7] = [-3,-3,-7]. Then you find the cross product of these two vectors as this will give a vector orthogonal to the plane i.e. the normal vector. In this case it is [-18,30,12]. All elements in this vector are a multiple of six, so we can rewrite the normal vector as [2,5,-3] to make it simpler. Therefore the equation of the plane is 2x + 5y - 3z = d. To find d, simply sub in one of the points given at the start which results in d = 21 = 2x + 5y -3z