Plane¶

Definition¶

A plane is a two-dimensional geometric object described by a point $\boldsymbol{A}$ and two linearly independent direction vectors $\boldsymbol{v},\boldsymbol{w}$ as $\boldsymbol{P}(s,t)=\boldsymbol{A}+s\boldsymbol{v}+t\boldsymbol{w}$$t\mathbf{w}$.

Interpretation¶

A plane is a flat two-dimensional surface: from a starting point one can move in two independent directions, and every point is reached by choosing how far to go in each. The parameters $s$ and $t$ are these two amounts, and the displacement $s\boldsymbol{v}+t\boldsymbol{w}$ is a linear combination of the direction vectors.

In the figure the plane passes through $\boldsymbol{A}$ and contains the directions $\boldsymbol{v}$ and $\boldsymbol{w}$. The choice $s=t=1$ gives the corner $\boldsymbol{P}(1,1)=\boldsymbol{A}+\boldsymbol{v}+\boldsymbol{w}$ of the dashed parallelogram. Other values of $s$ and $t$, negative ones included, fill the rest of the plane.

The parametrization needs linearly independent $\boldsymbol{v}$ and $\boldsymbol{w}$. If the directions are linearly dependent, the formula describes at most a line; if both are zero, it gives only the point $\boldsymbol{A}$. The description is not unique: $\boldsymbol{A}$ may be any point of the plane, and $\boldsymbol{v},\boldsymbol{w}$ any two independent direction vectors parallel to it. A plane through the origin is the span of $\boldsymbol{v}$ and $\boldsymbol{w}$, a two-dimensional subspace; any other plane is such a subspace shifted by $\boldsymbol{A}$.

In $\mathbb{R}^3$ a plane can also be described by a single equation, the point-normal equation with a normal vector. In $\mathbb{R}^n$, the one-equation description gives a hyperplane.

A common slip is to read $\boldsymbol{v}$ and $\boldsymbol{w}$ as points of the plane. They are directions: $\boldsymbol{A}+\boldsymbol{v}$ and $\boldsymbol{A}+\boldsymbol{w}$ lie on the plane, while $\boldsymbol{v}$ itself does so only when the plane passes through the origin.

Key Results¶

• Three non-collinear points determine a plane.
• If two direction vectors are scalar multiples, they do not determine a unique plane.

Examples¶

• $\boldsymbol{P}(s,t)=(0,0,0)+s(1,0,0)+t(0,1,0)$ is the $xy$-plane.
• The plane through three non-collinear points $\boldsymbol{A},\boldsymbol{B},\boldsymbol{C}$ can be written $\boldsymbol{P}(s,t)=\boldsymbol{A}+s(\boldsymbol{B}-\boldsymbol{A})+t(\boldsymbol{C}-\boldsymbol{A})$$\mathbf{A})$.

Prerequisites¶

• Linear Dependence and Independence
• Linear Combination
• Vector

Related Concepts¶

• Parametric Representations of Lines and Planes
• Hyperplane
• Affine Map
• Skew Lines

Sources¶

• Math II: Linear Algebra · Week 5

Keywords: affine plane, two-dimensional flat Symbols: real numbers

Last updated: 27 October 2026 11:52