Equation of a plane

A crystal is composed of atoms, molecules or ions that are arranged in a repetitive manner, forming a three-dimensional space lattice. Each point in the space lattice represents either a particle or a collection of particles (known as a basis), and a set of lattice points can be connected by a plane. As there are many possible ways to connect lattice points with planes of different orientations, a system is needed to define these planes. We begin by deriving the vector and scalar equations of a plane.

P(x, y, z) and P0(x0, y0, z0) are two points on a plane with position vectors r and r0 respectively, which makes – r0 the vector from Pto P. The plane has a direction defined by the normal vector n(A, B, C), which is perpendicular to the vector – r0. Therefore,

\textbf{\textit{n}}\cdot (\textbf{\textit{r}}-\textbf{\textit{r}}_0)=0\; \; \; \; \; \; \; (1)

\begin{pmatrix} A &B &C \end{pmatrix}\begin{pmatrix} x-x_0\\y-y_0 \\z-z_0 \end{pmatrix}=0



where D = Ax0 + By0 + Cz0.

\frac{x}{D/A}+\frac{y}{D/B}+\frac{z}{D/C}=1\; \; \; \; \; \; \; (2)

Eq1 is the vector equation of a plane, while eq2 is the scalar equation of a plane, where D/A, D/B and D/C are the intercepts of the plane with the x-axis, y-axis and z-axis respectively.


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