Using transformation formula and normal vector to create transformation matrix

Julianvn

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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 n=[n1n2n3]\overrightarrow{n} = \begin{bmatrix} n_1 \\ n_2 \\ n_3 \end{bmatrix}, using the formula fn(x)=x2(xnnnn)f_n(x) = x-2 \left( \frac{x \cdot n}{n \cdot n} \,\cdot\,n\right) leads to the matrix:

Fn=1n12+n22+n32[n12+n22+n322n1n22n1n32n2n1n12n22+n322n2n32n3n12n3n2n12+n22n32]\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}
 

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I have spent a lot of time on this question
Hello. Please show something of what you've tried, so that people can get a sense of what you already know. Thank you! ?

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Have you tried to express the formula for fn(x)f_n(x) through the coordinates of xx and nn ?
 
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