Plane

A plane is a flat two-dimensional surface that extends infinitely into all directions. There are many different ways to represent a plane. In multivariable calculus, planes are usually represented in scalar form; that is,

$ax+by+cz=k$

or

$z=Ax+By+k$

In linear algebra, planes are usually represented in vector notation. This can be done in two ways. The first is to define a point on the plane and the vector normal to the plane; for example,

$P_{0}={\begin{pmatrix}x_{0}\\y_{0}\\z_{0}\end{pmatrix}},{\vec {n}}={\begin{pmatrix}a\\b\\c\end{pmatrix}}$

The scalar equation of this plane will be

$a(x-x_{0})+b(y-y_{0})+c(z-z_{0})=0$

A plane can also be defined by a point and two non-parallel vectors

${\begin{pmatrix}x\\y\\z\end{pmatrix}}={\begin{pmatrix}x_{0}\\y_{0}\\z_{0}\end{pmatrix}}+t{\begin{pmatrix}x_{1}\\y_{1}\\z_{1}\end{pmatrix}}+s{\begin{pmatrix}x_{2}\\y_{2}\\z_{2}\end{pmatrix}}$