The equation of a plane in Cartesian coordinates is given by Ax + By + Cz + D = 0, where A, B, and C are the coefficients of the plane's normal vector and D is the distance from the origin along the normal vector.
To find the normal vector of a plane, you can use the cross product of two vectors that lie on the plane. The resulting vector will be perpendicular to both of the original vectors and thus will be the normal vector of the plane.
Yes, the equation of a plane can also be written in vector form as r · n = p, where r is a position vector, n is the normal vector, and p is a constant.
Three non-collinear points are needed to uniquely define a plane. This is because three points determine a plane and the normal vector of the plane can be found using the cross product of two vectors formed by these three points.
Calculus is used in finding the equation of a plane by using the concept of partial derivatives. By taking partial derivatives of the equation of the plane with respect to x, y, and z, the coefficients A, B, and C can be found, which are essential in defining the equation of the plane.