If each coordinate is distributed exponentially with parameter $\lambda$, then there are explicit expressions for the distribution of the inner product. E.g.:
In two dimensions where $\vec{a}=(a,b)$: $$P[\langle \vec{a},\vec{x} \rangle < d] = \frac{f(a)-f(b)}{a-b}\ \text{ with }\ f(x) = \max(x,0)-|x|e^{\min(0,-\lambda d/x)} $$
In three dimensions where $\vec{a}=(a,b,c)$ with $a,b,c,d>0$:
$$P[\langle \vec{a},\vec{x} \rangle < d] = \frac{f(a)}{(c-a)(a-b)}+\frac{f(b)}{(a-b)(b-c)}+\frac{f(c)}{(b-c)(c-a)} $$ $$\text{ with } g(x)=x^2 e^{-\lambda d/x}(1-e^{-\lambda d/x})$$