91影视

Let us consider $x_1-2x_2+2x_3=0$is an equation of a plane in ${\mathrm{鈩�}}^3$.

In the reflection, the basis of the plane consists of two vectors parallel to the plane and one vector perpendicular to the plane.

i.e. if$\mathfrak{I}=\left\{\stackrel{鈫�}{u},\stackrel{鈫�}{v},\stackrel{鈫�}{w}\right\}$ is the basis of the plane such thatrole="math" localid="1660375681690" $\stackrel{鈫�}{u},\stackrel{鈫�}{v}$ are parallel to the plane and$\stackrel{鈫�}{w}$ is perpendicular to the plane, then we have$\stackrel{鈫�}{w}=\stackrel{鈫�}{u}脳\stackrel{鈫�}{v}$ .

Since$\mathfrak{I}=\left\{\stackrel{鈫�}{u},\stackrel{鈫�}{v},\stackrel{鈫�}{w}\right\}$ is the basis of the plane such that$\stackrel{鈫�}{u},\stackrel{鈫�}{v}$ are parallel to the plane and is perpendicular to the plane, then the linear transformation of the vectors$\stackrel{鈫�}{u},\stackrel{鈫�}{v}$ and$\stackrel{鈫�}{w}$ is given by

$\begin{array}{l}T\left(\stackrel{鈫�}{u}\right)=\stackrel{鈫�}{u}\\ T\left(\stackrel{鈫�}{V}\right)=\stackrel{鈫�}{v}\\ T\left(\stackrel{鈫�}{W}\right)=-\stackrel{鈫�}{w}\end{array}$

Therefore the matrix $T=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& -1\end{array}\right]$is the reflection about the plane

$x_1-2x_2+2x_3=0$in ${\mathrm{鈩�}}^3$ .

If a 3 x 3 matrix T represents the reflection about a plane$x_1-2x_2+2x_3=0$ , then reflection$T=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& -1\end{array}\right]$ .