Find the axis of rotation of the composition L1 ◦ L2

Question

I'm having difficulty with this question from my Linear Algebra course,

I don't know how to answer the following,

Find the axis of rotation of the composition L1 ◦ L2

Answer 1

What you're looking for is the eigenvector associated with $\lambda = 1$ for the composition $L_1 \circ L_2$. In other words, you want a non-zero vector $v$ which will solve the equation $L_1(L_2(v)) = v$. This system can be routinely solved using the matrices for the transformations $L_2$ and $L_1$.

To think about it in an intuitive way, we want a vector for which the rotation about $e_3$ moves $v$ to $L_2(v)$, and the rotation about $e_1$ moves $L_2v$ back to $v$.

As it turns out, the correct vector to choose is $$ v = (1,1,-1) $$ (or any non-zero multiple of this vector). Verify that $$ L_2(v) = (1,-1,-1)\\ L_1(L_2(v)) = L_1(1,-1,-1) = (1,1,-1) $$