I want to find the PDF $f_\boldsymbol{X}(\vec{x})$ of a random variable $\boldsymbol{X}\in\mathbb{R}^3$ with positivly normally distributed length $|\boldsymbol{X}|=|x|$ where $x\sim\mathcal{N}(0,\sigma^2)$ and a unit direction (in the unit sphere $S^3$), i.e. $\hat{\boldsymbol{X}} = \boldsymbol{X}|\boldsymbol{X}|^{-1}\sim\mathcal{U}(S^3)$.
Since $\boldsymbol{X}$ results from the trivial composition $\boldsymbol{X}=|\boldsymbol{X}|\hat{\boldsymbol{X}}$, it should be possible to construct $f_\boldsymbol{X}$ from the normal and uniform distribution. A similar approach is used for for different compositions and PDFs, e.g. here or here.
My attempt was the following: Let $\omega\in\mathbb{R}^3$. We know that $$ Pr(\hat{\boldsymbol{X}}\in\omega) = \frac{|\omega ~\cap~S^3|}{|S^3|} $$ and thus $$ Pr(\boldsymbol{X}\in\omega) = Pr(\hat{\boldsymbol{X}} \in \omega|\boldsymbol{X}|^{-1}). $$ Now my most intuitive next step is $$ Pr(\hat{\boldsymbol{X}} \in \omega|\boldsymbol{X}|^{-1}) =\frac{|\omega |\boldsymbol{X}|^{-1}~\cap~S^3|}{|S^3|} $$ which clearly is non-sense, since the random variable $|\boldsymbol{X}|^{-1}$ apprears in the CDF.
How do I go on from here? I would very much like to solve this using first principles and bare probabilities.
Let $$X = R \begin{bmatrix} \sin\theta \cos\phi \\ \sin\theta \sin\phi \\ \cos \phi \end{bmatrix}$$ Then if you choose $U, V \sim Uni(0, 1)$ i.i.d, you can get a uniform distribuiton on the sphere R, by using: $$\theta = 2 \pi u$$ $$\phi = \arccos (2v - 1)$$
So what you want is $$U, V \sim Uni(0, 1)$$ $$R \sim |N(0, \sigma^2)|$$ $$\theta = 2 \pi u$$ $$\phi = \arccos (2v - 1)$$
For more informationon this, see: https://mathworld.wolfram.com/SpherePointPicking.html