In R3\mathbb{R}^3R3, find the matrix for a rotation of 180∘180^\circ180∘ about the line spanned by (1,1,1)(1, 1, 1)(1,1,1).
13(−1222−1222−1)\frac{1}{3} \begin{pmatrix} -1 & 2 & 2 \\ 2 & -1 & 2 \\ 2 & 2 & -1 \end{pmatrix}31−1222−1222−1
(010100001)\begin{pmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{pmatrix}010100001
13(122212221)\frac{1}{3} \begin{pmatrix} 1 & 2 & 2 \\ 2 & 1 & 2 \\ 2 & 2 & 1 \end{pmatrix}31122212221
(−1000−1000−1)\begin{pmatrix} -1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1 \end{pmatrix}−1000−1000−1