Let $x,y\in\mathbb{R}^d$ be i.i.d. $\mathcal{N}(0,I_d)$ distributed, and let $a,b\in\mathbb{R}$ be arbitrary. Then are the random variables
$$ \|x\|_2 \quad\text{and}\quad \frac{ax+by}{\|ax+by\|_2} $$
independent?
I feel like the answer should be yes because revealing the magnitude gives no information regarding the direction, but I am not very confident in writing down derivations for verifying it. Thank you!