# Find the matrix of ortogonal reflection

Let $e_1, e_2, e_3$ be an orthonormal basis for $R^3$ and consider the plane with equation $x_1 + 2x_2 - 2x_3 = 0$. Find the matrix of orthogonal reflection in that plane with respect to the given basis.

So, first I know that an orthogonal reflection satisfies: 1. F is a reflection 2. F is symmetric

So if I could find a transformation matrix, I would easily be able to verify that it is a reflection by the shape of the matrix and by asserting that $A^2 = I$. However, I don't know how to find such a matrix.

• The equation of the plane is in the standard basis or in the basis $\{e_i\}$? – Emilio Novati Apr 9 '16 at 9:08
• I guess that $e_1, e_2, e_3$ is defined as the standard basis, i.e. $e_1 = (1,0,0)...$. – Ludwwwig Apr 9 '16 at 9:20

I suppose that ${e_i}$ is the standard basis. In this case you can find the matrix from a geometrical construction.
If $\vec u$ is the vector orthogonal to the plane, the projection of a vector $x=[x_1,x_2,x_3]^T$ on $\vec u$ is: $$\vec v= \frac {\langle \vec x, \vec u \rangle}{|u|^2}\vec u$$ and the reflection of $\vec x$ in the plane gives a vector
$$\vec x'=\vec x - 2 \vec v$$
In your case $\vec u=[1,2,-2]^T$. can you do from this?