- #1
Irid
- 207
- 1
In Goldstein there is a problem asking to find a vector representation for a reflection in a plane of a unit normal \mathbf{\hat{n}}. I find it to be
\mathbf{r'} = \mathbf{r} - 2(\mathbf{r\cdot \hat{n}})\mathbf{\hat{n}}
and it has a corresponding transformation matrix with elements
A_{ij} = \delta_{ij} - 2l_i l_j
where l_i\, , i=1,2,3 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 A^T = A, and then I multiply A^2 = I, which shows that A^T = A^{-1}, 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:
\text{det}(A) = \begin{vmatrix}<br /> 1-l_1^2 & -l_1 l_2 & -l_1 l_3\\<br /> -2l_1 l_2 & 1-2l_2^2 & -2l_2 l_3\\<br /> -2l_1 l_3 & -2l_2 l_3 & 1-2l_3^2\end{vmatrix} = 1
Does it mean that the matrix is a proper one? Or is there an error in the problem statement, or am I missing something?
\mathbf{r'} = \mathbf{r} - 2(\mathbf{r\cdot \hat{n}})\mathbf{\hat{n}}
and it has a corresponding transformation matrix with elements
A_{ij} = \delta_{ij} - 2l_i l_j
where l_i\, , i=1,2,3 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 A^T = A, and then I multiply A^2 = I, which shows that A^T = A^{-1}, 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:
\text{det}(A) = \begin{vmatrix}<br /> 1-l_1^2 & -l_1 l_2 & -l_1 l_3\\<br /> -2l_1 l_2 & 1-2l_2^2 & -2l_2 l_3\\<br /> -2l_1 l_3 & -2l_2 l_3 & 1-2l_3^2\end{vmatrix} = 1
Does it mean that the matrix is a proper one? Or is there an error in the problem statement, or am I missing something?
