Finding a rotation matrix

Question

I am looking for a rotation matrix such that $$ \operatorname{rot} \cdot \begin{pmatrix} -1 & 0 & 0 \\ 1 & -1 & 0 \\ 0 & 1 & -1 \\ 0 & 0 & 1 \end{pmatrix} = \begin{pmatrix} \times & \times & \times \\ \times & \times & \times \\ \times & \times & \times \\ 0 & 0 & 0 \end{pmatrix}. $$ I think this should be possible because the columns of my matrix are all in the $3$-dimensional hyperplane $\{ (x_1, x_2, x_3, x_4) \in \mathbb{R}^4 : x_1 + x_2 + x_3 + x_4 = 0 \} \subset \mathbb{R}^4$.

Answer 1

Considering the bottom entry of your "rot," you want its dot product with each column to be zero. That can only happen if the entries are all the same, so the bottom row of rot is $\pm\big(\frac 1 2\ \frac 1 2\ \frac 1 2\ \frac 1 2\big)$.

Once you know that, you just need to find three more orthogonal unit vectors.

Answer 2

Hint: consider a rotation such as $$ R(1,1,1,1) = (0,0,0,1) $$