Find the image of the point $(-2,1,1)$ in the plane $x+y+z=0$.

The formula for the image $(x^{\prime },y^{\prime },z^{\prime })$ of point $(x_1,y_1,z_1)$ in plane $ax+by+cz+d=0$ is:
$$\frac{x^{\prime }-x_1}{a}=\frac{y^{\prime }-y_1}{b}=\frac{z^{\prime }-z_1}{c}=-2\frac{a{x}_1+b{y}_1+c{z}_1+d}{a^2+b^2+c^2}$$
Here $(x_1,y_1,z_1)=(-2,1,1)$ and plane $1x+1y+1z=0$ ($d=0$).Calculate the ratio $R$:
$$R=-2\frac{(1)(-2)+(1)(1)+(1)(1)}{1^2+1^2+1^2}=-2\frac{-2+1+1}{3}=-2\frac{0}{3}=0$$
Since $R=0$:
$$x^{\prime }-(-2)=0⟹x^{\prime }=-2$$
$$y^{\prime }-1=0⟹y^{\prime }=1$$
$$z^{\prime }-1=0⟹z^{\prime }=1$$
The point $(-2,1,1)$ lies on the plane, so its image is itself.