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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}

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