Characteristic function of Multi dimensional Brownian motion stopped at hitting time

Question

I am trying to solve the following question: denote $(X_t,Y_t)$ a Brownian motion in $\mathbb{R}^n\times\mathbb{R}$ starting from $(0,a)$ with $a>0$. Let $T_a=\inf\{t:Y_t=0 \}$. Prove that the characteristic function of $X_{T_a}$ is $\exp(-a \Vert u\Vert)$. All I can prove is that by independence, $$\mathbb{E}[e^{ i\langle u,X_{T_a}\rangle}]=\mathbb{E}[e^{ i\sum_j u_jX^j_{T_a}}]=\mathbb{E}[\prod_je^{ i u_jX^j_{T_a}}]=\prod_j\mathbb{E}[e^{ i u_jX^j_{T_a}}]=\prod_je^{-a|u_j|}=e^{-a\sum_j|u_j|}$$ And obviously, $$e^{-a\sum_j|u_j|}\neq e^{-a \Vert u\Vert}$$ What am I doing wrong ?