Reflection of a plane in a plane.

Question

The question is:

The reflection of the plane $2x+3y+4z-3=0$ in the plane $x-y+z-3=0$ is the plane:

I tried to find the equation of the normal to the plane and then tried putting in some values, but I couldn't do it. I know how to find the reflection of a point about a plane, but I have no idea how to proceed in this one.

Thanks.

Answer 1

Start with reflecting the normal of $2x+3y+4z-3=0$ in $x-y+z-3=0$ to get a new normal vector $(a,b,c)$. Write an equation $ax+by+cz+d=0$ for the new plane, and reflect a point of $2x+3y+4z-3=0$ in $x-y+z-3=0$ to get a point $(x,y,z)$. Insert this point into $ax+by+cz+d=0$ to find out $d$.

Answer 2

I propose this strategy ( but I don't know if it's the simpler):

1) find a translation that brings the given planes to parallel planes passing through the origin. This is done taking a point on the straight line common to the two planes. If I've not made some mistake we can take the translation $ T(x,y,z)\rightarrow \left(x,y+\dfrac{15}{7},z-\dfrac{6}{7}\right) $ and we have the two planes: $$ a)\qquad 2x+3y+4z=0 $$ $$ b)\qquad x-y+z=0 $$ the first orthogonal to: $\mathbf{v}=(2,3,4)^T$ and the second orthogonal to: $\mathbf{u}=(1,-1,1)^T$

2) Now the reflection of $\mathbf{v}$ through the plane $b)$ is given by (see here): $$ R(\mathbf{v})=\mathbf{v}-2\dfrac{\langle\mathbf{v},\mathbf{u} \rangle}{\langle\mathbf{u},\mathbf{u}\rangle}\mathbf{u} =(v_1',v_2',v_3')^T $$ and the reflected plane has equation $$ v_1'x+v_2'y+v_3'z=0 $$ 3) use the inverse translation $T^{-1}(x,y,z)\rightarrow \left(x,y-\dfrac{15}{7},z+\dfrac{6}{7}\right) $ to find the request plane.

Answer 3

The most basic, easy to understand and straightforward method according to me is this:-

1. Consider any point on the plane that has to be reflected, here $P_1:2x+3y+4z−3=0$. Let $(0,0,\frac 34)$ (or any other of your choice)

2. Now find the reflection of the point in the mirror plane, here $P_2:x−y+z−3=0$.

3. We know that any plane in space is a linear combination of two distinct planes, let the reflected plane is $P_3=P_1 +λP_2$. As the reflection lies in the reflected plane, put the coordinates of the reflected point in $P_3$ and get $λ$, hence $P_3$.