Intercept Form of a Plane

Plane making intercepts p, q, r on the x, y, z axes respectively. Passes through (p, 0, 0), (0, q, 0), (0, 0, r).

A plane makes intercepts $p$, $q$, $r$ on the x, y, z axes — so it passes through $(p,0,0)$, $(0,q,0)$, $(0,0,r)$.

Let the plane be $ax+by+cz = d$. Substituting each intercept:

$$ap = d \implies a = d/p, \quad bq = d \implies b = d/q, \quad cr = d \implies c = d/r$$

Substituting into $ax+by+cz = d$ and dividing by $d$:

$$\frac{x}{p}+\frac{y}{q}+\frac{z}{r} = 1$$

When an intercept is infinite: The plane is parallel to that axis. For example, if $r \to \infty$, the $z/r$ term vanishes and the plane becomes $x/p+y/q=1$ (parallel to z-axis).

Normal direction: From the general form $x/p+y/q+z/r=1$, the normal has DRs $(1/p, 1/q, 1/r)$ or equivalently $(qr, pr, pq)$.