Relationship between the solution of $\arg\!\min_a |\frac{x}{|x|_1} - ay|_1$ and that of $\arg\!\min_b |\frac{y}{|y|_1} - bx|_1$

Question

Here $x, y$ are non-negative real vectors, $|x|_1 \neq 0, |y|_1 \neq 0$ and $a, b$ are real scalars. I am thinking that

$$ \arg\!\min_a |\frac{x}{|x|_1} - ay|_1 = \arg\!\min_a |x - a|x|_1y|_1$$

and similarly \begin{align*} \arg\!\min_b |\frac{y}{|y|_1} - bx|_1 = \arg\!\min_b |bx - \frac{y}{|y|_1}|_1 \\ \neq \arg\!\min_b |x - \frac{y}{b|y|_1}|_1 \end{align*}

If I solve for one of them, can I somehow algebraically get the solution of the other one?