I have spent a lot of time on this question and I just cannot figure it out. I have asked peers and no one knows how to do it.
Show that, for the normal vector [imath]\overrightarrow{n} = \begin{bmatrix} n_1 \\ n_2 \\ n_3 \end{bmatrix}[/imath], using the formula [imath]f_n(x) = x-2 \left( \frac{x \cdot n}{n \cdot n} \,\cdot\,n\right)[/imath] leads to the matrix:
[imath]\qquad \qquad F_{\overrightarrow{n}} = \frac{1}{n_1^2 + n_2^2 + n_3^2} \begin{bmatrix} -n_1^2 + n_2^2 + n_3^2 & -2n_1 n_2 & -2n_1 n_3 \\ -2n_2n_1 & n_1^2 - n_2^2 + n_3^2 & -2n_2 n_3 \\ -2n_3 n_1 & -2n_3 n_2 & n_1^2 + n_2^2 - n_3^2 \end{bmatrix}[/imath]
Show that, for the normal vector [imath]\overrightarrow{n} = \begin{bmatrix} n_1 \\ n_2 \\ n_3 \end{bmatrix}[/imath], using the formula [imath]f_n(x) = x-2 \left( \frac{x \cdot n}{n \cdot n} \,\cdot\,n\right)[/imath] leads to the matrix:
[imath]\qquad \qquad F_{\overrightarrow{n}} = \frac{1}{n_1^2 + n_2^2 + n_3^2} \begin{bmatrix} -n_1^2 + n_2^2 + n_3^2 & -2n_1 n_2 & -2n_1 n_3 \\ -2n_2n_1 & n_1^2 - n_2^2 + n_3^2 & -2n_2 n_3 \\ -2n_3 n_1 & -2n_3 n_2 & n_1^2 + n_2^2 - n_3^2 \end{bmatrix}[/imath]
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