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 $\vec{a}=(a,b)$: $$P[\langle \vec{a},\vec{x} \rangle < d] = \frac{f(a)-f(b)}{a-b}\ \text{ with }\ f(u) = \max(u,0)-|u|e^{\min(0,-\lambda d/u)} $$

In three dimensions where $\vec{a}=(a,b,c)$, with $a,b,c,d>0$:

$$P[\langle \vec{a},\vec{x} \rangle < d] = \frac{g(a)}{(c-a)(a-b)}+\frac{g(b)}{(a-b)(b-c)}+\frac{g(c)}{(b-c)(c-a)} $$ $$\text{ with } g(u)=u^2 e^{-\lambda d/u}(1-e^{-\lambda d/u})$$

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 $\vec{a}=(a,b)$: $$P[\langle \vec{a},\vec{x} \rangle < d] = \frac{f(a)-f(b)}{a-b}\ \text{ with }\ f(u) = \max(u,0)-|u|e^{\min(0,-\lambda d/u)} $$

In three dimensions where $\vec{a}=(a,b,c)$, with $a,b,c,d>0$:

$$P[\langle \vec{a},\vec{x} \rangle < d] = \frac{g(a)}{(c-a)(a-b)}+\frac{g(b)}{(a-b)(b-c)}+\frac{g(c)}{(b-c)(c-a)} $$ $$\text{ with } g(u)=u^2 (e^{-\lambda d/u}-1)$$

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 $a,b,c,d>0$:

$$P[\langle \vec{a},\vec{x} \rangle < d] = \frac{f(a)}{(c-a)(a-b)}+\frac{f(b)}{(a-b)(b-c)}+\frac{f(c)}{(b-c)(c-a)} $$ $$\text{ with } g(x)=x^2 e^{-\lambda d/x}(1-e^{-\lambda d/x})$$

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 $\vec{a}=(a,b)$: $$P[\langle \vec{a},\vec{x} \rangle < d] = \frac{f(a)-f(b)}{a-b}\ \text{ with }\ f(u) = \max(u,0)-|u|e^{\min(0,-\lambda d/u)} $$

In three dimensions where $\vec{a}=(a,b,c)$, with $a,b,c,d>0$:

$$P[\langle \vec{a},\vec{x} \rangle < d] = \frac{g(a)}{(c-a)(a-b)}+\frac{g(b)}{(a-b)(b-c)}+\frac{g(c)}{(b-c)(c-a)} $$ $$\text{ with } g(u)=u^2 e^{-\lambda d/u}(1-e^{-\lambda d/u})$$