A linear equation in 3 unknowns $ax + by + cz = d$ represents a plane in $\mathbb{R}^3$.
- If one coefficient is zero (e.g., $c = 0$), the plane is parallel to the corresponding axis
- If two coefficients are zero, the plane is parallel to a coordinate plane
Special cases:
- $ax + by + cz = 0$ — the plane passes through the origin
- $0x + 0y + 0z = 0$ — the entire space $\mathbb{R}^3$
- $0x + 0y + 0z = d$ (with $d \neq 0$) — the empty set
Example: $2x + 3y + z = 6$ is a plane in $\mathbb{R}^3$ passing through $(3, 0, 0)$, $(0, 2, 0)$, and $(0, 0, 6)$.
The geometric description assumes that each equation has at least one nonzero coefficient. An all-zero left side instead gives the whole ambient space when the constant is zero, or no solutions when it is nonzero.
Like how this is organised?
Build your own knowledge tree on knowen.md: notes, decisions and tasks in one connected markdown graph, shared with your team and your AI agents.