Math101 · Independent Ontario learning library
Math101Planes
A rigorous, example-driven guide to planes, including hypotheses, method choice, verification, and practice.
The central idea
A plane in $\mathbb R^3$ with nonzero normal $\mathbf n=\langle a,b,c\rangle$ through $P_0=(x_0,y_0,z_0)$ satisfies $\mathbf n\cdot(\mathbf r-\mathbf r_0)=0$, or $a(x-x_0)+b(y-y_0)+c(z-z_0)=0$. Expanded form is $ax+by+cz=d$. Two planes are parallel when their normals are parallel and perpendicular when their normals have dot product zero.
Definitions, hypotheses, and notation
Three noncollinear points determine a plane: subtract one base point from the other two and cross the resulting directions. Collinear points produce parallel differences and no unique plane. The angle between planes is conventionally the acute angle between normals, found from the absolute dot-product formula.
The distance from point $P$ to $ax+by+cz=d$ is $|aP_x+bP_y+cP_z-d|/\sqrt{a^2+b^2+c^2}$. Scaling the entire plane equation leaves this ratio unchanged, reinforcing that normal magnitude is not geometrically fixed.
Conceptual meaning
A plane contains every displacement from its base point perpendicular to one fixed normal. Two independent in-plane directions span it. Normal form is usually the safest representation because it exposes orientation and avoids solving for a missing coordinate.
A dependable method and decision rule
- Obtain a normal directly or cross two independent in-plane directions.
- Choose a verified point on the plane.
- Write point-normal form before expanding.
- Test points by substitution and line directions by dot product with the normal.
- Use normal comparisons for angles, parallelism, and intersections.
