Equation of a Plane in 3D: ax + by + cz = d

A plane in 3D is defined by the equation $ax + by + cz = d$. The vector $(a, b, c)$ is the normal vector — it's perpendicular to the plane. The intercepts are at $(d/a, 0, 0)$, $(0, d/b, 0)$, $(0, 0, d/c)$.

For $2x + 3y + z = 6$: normal = $(2, 3, 1)$, intercepts at $(3, 0, 0)$, $(0, 2, 0)$, $(0, 0, 6)$.

Try it in the visualization

Adjust the $a$, $b$, $c$, $d$ sliders and watch the plane tilt and shift. The normal vector arrow shows which direction the plane faces. Toggle intercepts to see where it crosses each axis.