The discussion focuses on deriving the Cartesian equation of a plane through three given points: (1,0,3), (2,-4,3), and (4,-1,2). To find the equation, two vectors are created from these points, and their cross product is calculated to determine the normal vector of the plane. The final equation is expressed in the form A(x - x_0) + B(y - y_0) + C(z - z_0) = 0, where A, B, and C are components of the normal vector, and (x_0, y_0, z_0) is one of the points on the plane. The solution was confirmed by the original poster after initial confusion regarding the process.
PREREQUISITESStudents studying geometry, particularly those learning about planes in three-dimensional space, as well as educators looking for examples to illustrate vector operations and plane equations.
Use two pairs of those points to get two vectors in the plane: the vector from (x_0, y_0, z_0) to (x_1, y_1, z_1) is (x_1- x_0)\vec{i}+ (y_1- y_0)\vec{j}+ (z_1- z_0)\vec{k}.HallsofIvy said:Haven't been paying attention in class?
The cross product of two vectors is perpendicular to the two vectors so perpendicular to the plane they lie in.
The equation of a plane with normal vector A\vec{i}+ B\vec{j}+ C\vec{k} containing point (x_0, y_0, z_0) is A(x- x_0)+ B(y- y_0)+ C(z- z_0)= 0.