Matrix Reflection Homework: Find Orthogonal Matrix in R3 Plane

[DanZ123]

Homework Statement

Let u1,u2,u3 be an orthonormal basis for R3 and consider M as the plane with equation x1+2x2-2x3=0. Find the matrix of orthogonal reflection in that plane with respect to the given basis.

Homework Equations

The Attempt at a Solution

In previous exercises , I had a matrix A given and was asked to find the equation of the plane that the matrix was projected or reflected on. To do that I solved the equation (A-I)x=0 . (The nullspace/kernel minus the identity matrix) ..
I was thinking that maybe to solve this current exercise, I could maybe use the method from the previous exercises but use it backwards ... and find the matrix A?
But I am not sure if I am thinking right, or how to attack the problem...
Would appreciate help...thanks.

 

[geoffrey159]

The direction of your reflection is $M^\perp =\mathbb{R} (1,2,-2)$. Find an orthonormal basis of $\mathbb{R}^3$ starting with a unit vector of $M^\perp$. With respect to that basis, the matrix of reflection is $\begin{pmatrix} -1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}$. What do you need to complete the exercise ?

 

[HallsofIvy]

Homework Statement

Let u1,u2,u3 be an orthonormal basis for R3 and consider M as the plane with equation x1+2x2-2x3=0. Find the matrix of orthogonal reflection in that plane with respect to the given basis.

This doesn't quite make sense. It would be easy to find the "matrix of orthogonal reflection in x1+ 2x2- x3= 0 in the standard basis i, j, k, but what is the relation of u1, u2, u3 to that?

2. Homework Equations

The Attempt at a Solution

In previous exercises , I had a matrix A given and was asked to find the equation of the plane that the matrix was projected or reflected on. To do that I solved the equation (A-I)x=0 . (The nullspace/kernel minus the identity matrix) ..
I was thinking that maybe to solve this current exercise, I could maybe use the method from the previous exercises but use it backwards ... and find the matrix A?
But I am not sure if I am thinking right, or how to attack the problem...
Would appreciate help...thanks.

 

[DanZ123]

This doesn't quite make sense. It would be easy to find the "matrix of orthogonal reflection in x1+ 2x2- x3= 0 in the standard basis i, j, k, but what is the relation of u1, u2, u3 to that?

Ok so how do you find it for the standard basis i,j,k?

 

[DanZ123]

The direction of your reflection is $M^\perp =\mathbb{R} (1,2,-2)$. Find an orthonormal basis of $\mathbb{R}^3$ starting with a unit vector of $M^\perp$. With respect to that basis, the matrix of reflection is $\begin{pmatrix} -1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}$. What do you need to complete the exercise ?

Im not sure that iam following what's happening...

 

[geoffrey159]

A reflection is an orthogonal symetry with respect to a plane.

Its direction $\Delta$ is the vector line orthogonal to the plane, and given a cartesian equation of the plane of symetry, you have immediate access to an orthogonal vector to that plane, don't you ?

Without going into the details, you understand with a sketch that an orthogonal symetry is stable in the plane of symetry and transforms vectors of $\Delta$ into their opposite.

Now given an orthonormal basis of $\mathbb{R}^3$ starting with a unit vector of $\Delta$, the two other vectors belong to the plane of symetry, don't they ? What is the matrix of your reflection in that basis ?

Now you want to express this matrix into another basis, the canonical basis of $\mathbb{R}^3$, so you need a change of basis matrix.