Equations of planes in 3D

The equation of a plane in three dimensions (3D) is given by Ax + By + Cz = D, where A, B, C, and D are constants, and x, y, and z are variables. This is referred to as the standard form of the equation of a plane.
Each of the coefficients A, B, and C indicates the direction ratios of the normal to the plane. The normal is a vector perpendicular to the plane.
The formula can also be written in vector form as r.n = d, where r is the position vector (x, y, z), n is the normal vector (A, B, C), and d is a constant equal to the displacement of the plane from the origin along the normal vector.
One method to find the equation of a plane is by using three non-collinear (not in a straight line) points. Firstly, two direction vectors are found by subtracting the coordinates of the points. The cross product of these direction vectors then gives the normal to the plane.
When a plane intercepts the x, y, and z axes, the equation can be framed as x/a + y/b + z/c = 1 where a, b, and c represent the points where the plane intercepts the axes. This is sometimes easier to visualize.
A plane is parallel to another if their normal vectors are multiples of each other. For instance, if the normals are n1 and n2, then n1=kn2, where k is a constant.
The distance d from a point P(x1, y1, z1) to a plane Ax + By + Cz = D is given by the formula d = Ax1 + By1 + Cz1 - D / sqrt(A^2 + B^2 + C^2), where sqrt indicates the square root.
Two planes are perpendicular if the dot product of their normal vectors equals zero. So if the normals are n1 and n2, then n1.n2 = 0 implies that the planes are perpendicular.
The angle θ between two planes can be found using the dot product rule between their normal vectors: cos(θ) = n1.n2 / n1 n2 .
When solving sets of three planes, one could either use substitution or elimination, much like how you would solve simultaneous equations. Alternatively, when three planes intersect at a single point, it may be necessary to utilize matrices and determinant to solve.
Practice on past questions, textbook problems, and online quizzes will focus your understanding of various question types and strengthen conceptual understanding. Always look out for links with other topics to build a holistic understanding of Pure Mathematics.