In Goldstein there is a problem asking to find a vector representation for a reflection in a plane of a unit normal [tex]\mathbf{\hat{n}}[/tex]. I find it to be(adsbygoogle = window.adsbygoogle || []).push({});

[tex] \mathbf{r'} = \mathbf{r} - 2(\mathbf{r\cdot \hat{n}})\mathbf{\hat{n}}[/tex]

and it has a corresponding transformation matrix with elements

[tex]A_{ij} = \delta_{ij} - 2l_i l_j[/tex]

where [tex]l_i\, , i=1,2,3[/tex] are the direction cosines for the orientation of the plane. Goldstein then asks to show that this matrix is an improper orthogonal one. I can show orthogonality by simply noting that [tex]A^T = A[/tex], and then I multiply [tex]A^2 = I[/tex], which shows that [tex]A^T = A^{-1}[/tex], which is the condition for orthogonality.

However, the improper nature of the matrix is unclear to me. If I compute the determinant, by explicitly writing out the form of the matrix, the result I obtain is +1, instead of -1:

[tex] \text{det}(A) = \begin{vmatrix}

1-l_1^2 & -l_1 l_2 & -l_1 l_3\\

-2l_1 l_2 & 1-2l_2^2 & -2l_2 l_3\\

-2l_1 l_3 & -2l_2 l_3 & 1-2l_3^2\end{vmatrix} = 1[/tex]

Does it mean that the matrix is a proper one? Or is there an error in the problem statement, or am I missing something?

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# Homework Help: Improper rotation matrix

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