Planes

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

1. Obtain a normal directly or cross two independent in-plane directions.
2. Choose a verified point on the plane.
3. Write point-normal form before expanding.
4. Test points by substitution and line directions by dot product with the normal.
5. Use normal comparisons for angles, parallelism, and intersections.

Fully worked example

Graphical or geometric meaning

The normal arrow pierces the plane at a right angle. Changing the constant $d$ while keeping $\langle a,b,c\rangle$ fixed slides the plane parallel to itself.

Common mistakes and why they fail

Verification and reasonableness checks

• Substitute the defining point.
• Dot the normal with known in-plane directions.
• Confirm the normal is nonzero and coefficient expansion is consistent.

The normal vector carries the plane

A plane through $\mathbf r_0$ with nonzero normal $\mathbf n$ satisfies $\mathbf n\cdot(\mathbf r-\mathbf r_0)=0$. Any nonzero scalar multiple of $\mathbf n$ describes the same plane. Three noncollinear points determine a normal through the cross product of two displacement vectors; a zero cross product signals that the chosen points are collinear or repeated. To test a candidate point, substitute it into the scalar equation. Parallel planes have parallel normals, while perpendicular planes have orthogonal normals. The distance formula divides the substituted plane expression by $\|\mathbf n\|$, so normalization cannot simply be ignored there. Distinguish a normal vector from a direction lying within the plane.

Practice

1. Find the plane through the origin normal to $\langle1,2,3\rangle$.
2. Are $x+y+z=1$ and $2x+2y+2z=5$ parallel?
3. What is a normal to $z=4$?

Connections and next steps

• Planes in Three Dimensions
• Lines in Space
• Tangent Planes