Planes in Three Dimensions

A plane in three-dimensional space is determined by a point and a normal vector perpendicular to every direction in the plane.

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A plane is controlled by a point and two directions, or equivalently by a normal vector. Three coordinate axes support a tilted plane patch. One perpendicular normal arrow distinguishes the plane’s position from its orientation.

A normal vector turns a flat surface in space into one linear equation.

Point-normal form

Let a plane pass through $P_0(x_0,y_0,z_0)$ with nonzero normal vector

\vec n=\langle a,b,c\rangle.

For any plane point $P(x,y,z)$, the displacement $\overrightarrow{P_0P}$ lies in the plane and is perpendicular to $\vec n$. Therefore

\vec n\cdot\overrightarrow{P_0P}=0,

$$ a(x-x_0)+b(y-y_0)+c(z-z_0)=0. $$

Worked example: point and normal

Vector and parametric forms

If $\vec u$ and $\vec v$ are nonparallel directions in the plane, then

\vec r=\vec r_0+s\vec u+t\vec v,qquad s,t\in\mathbb R.

Two independent parameters sweep across a surface. The normal is perpendicular to both direction vectors.

Relationships between planes

Planes are parallel when their normal vectors are scalar multiples. They are coincident if they are parallel and share a point or have proportional complete scalar equations.

Planes are perpendicular when their normal vectors are perpendicular:

\vec n_1\cdot\vec n_2=0.

Two nonparallel planes intersect in a line.

Common mistakes

Using an in-plane direction as the normal. The normal must dot to zero with every plane direction.

Forgetting that proportional normals can describe parallel planes. Compare full equations to decide coincidence.

Using only two points to define a unique plane. Infinitely many planes contain one line.

Taking a cross product of collinear directions and accepting the zero vector as a normal. A normal must be nonzero.

Comparing plane angles without absolute value. Opposite normals represent the same orientation.

Quick self-check

Is the normal vector nonzero and perpendicular to known plane directions?
Does the equation contain the supplied point?
Are equivalent scalar multiples recognized?
If three points are used, are they noncollinear?
Do plane relationships follow from their normals?
Are intercepts, distance, and units interpreted correctly?