{title2=$P$}

A plane is a two-dimensional <affine subspace>: a point $\mathbf a$ together with every <linear combination> of two <linearly independent> direction <vectors> $\mathbf u,\mathbf v$,
$$
P=\{\mathbf a+s\mathbf u+t\mathbf v:s,t\in\mathbb R\}.
$$
In three-dimensional <Euclidean space>, a <Cartesian equation of a plane> fixes one <linear equation> $\mathbf n\cdot\mathbf x=d$, with a nonzero <normal vector> $\mathbf n$. The two free directions are <orthogonal> to $\mathbf n$. A <plane> extends indefinitely; a rectangular patch in a drawing marks only part of it, and does not define an edge of the <plane>.

For example, the <plane> $2x-y+2z=3$ contains $\mathbf a=(1,1,1)$. The <vectors> $\mathbf u=(1,2,0)$ and $\mathbf v=(0,2,1)$ are <linearly independent> and satisfy $(2,-1,2)\cdot\mathbf u=(2,-1,2)\cdot\mathbf v=0$. Hence its points are exactly $(1+s,1+2s+2t,1+t)$: substitution gives $2x-y+2z=3$ for every real $s,t$, and every solution has $s=x-1$, $t=z-1$.

The description separates position from direction. A <translation> of the <plane> changes $\mathbf a$ and possibly $d$, while retaining its parallel directions. A <plane> through the <origin> is a <linear subspace>; the displayed example is not, since the <zero vector> fails its <linear equation>. This distinction matters when applying <orthogonal projection>: subtract an anchor point before projecting onto the direction <linear subspace>.

Historically, <Euclid>'s <Euclid's Elements by Euclid>, Book I Definition 7, describes a plane surface through the <straight lines> lying on it. <David E. Joyce>'s accompanying guide notes that this older use can include a bounded square or circular figure. That usage differs from the whole unbounded <affine subspace> used here.<Euclid's Elements by Euclid/Book I, Definition 7>[Euclid's Elements, Book I Definition 7 and David Joyce's Guide: plane surfaces and bounded figures]{ref}


 Plane (source code)