I need to sample a vector $(\gamma_1,\ldots,\gamma_k)$ which belongs to a hypersphere with radius $\varphi$ and to the hyperplane orthogonal to the $k$ dimensional vector $(1,\ldots,1)$. This can be found by
$$(\gamma_1,\ldots,\gamma_k)=\varphi\cos(\varpi_1)F_1+\varphi\sin(\varpi_1)\cos(\varpi_2)F_2+\ldots+\varphi\sin(\varpi_1)\ldots\sin(\varpi_{k-2})F_{k-1}$$
where $(F_1,\ldots,F_{k-1})$ is an orthonormal basis of the hyperplane, $(\varpi_1,\ldots,\varpi_{k-3})\sim\text{Unif}([0,\pi]^{k-3})$, and $\varpi_{k-2}\sim\text{Unif}([0,2\pi])$ Section 2.3.1.
The problem that I am having is that when $k$ is large, the higher dimensions of $(\gamma_1,\ldots,\gamma_k)$ become very small. For example, one such vector I just sampled for $k=48$ was:
[0.474436, -0.445468, -0.052532, 0.240382, -0.174982, -0.064045, 0.0124799, 0.0076257, 0.00622904, -0.00415307, -0.000411894, 0.000242749, 0.000130771, 7.44666e-05, -1.04834e-05, -1.69516e-06, 2.82631e-05, -3.47123e-05, 2.5018e-05, -9.31519e-06, 5.37815e-06, -9.37462e-06, -4.97792e-06, 2.94351e-06, 7.61421e-08, 1.24979e-08, -7.4654e-09, -6.9651e-10, -6.03173e-11, 3.22116e-11, -6.21856e-11, 1.7103e-11, -9.91226e-13, 9.74194e-13, 1.06403e-12, 4.49213e-13, 1.91792e-14, 3.27387e-15, -4.90812e-15, -6.11207e-17, 3.28663e-17, 1.95478e-16, -7.30665e-17, -3.30093e-17, 1.57977e-17, 2.06998e-17, -4.10022e-18, 7.92572e-18]
Presumably, this is due to repeated multiplications of $\sin(\cdot)$.
My questions is: is this occurring because of how I am sampling, or due to the nature of the sub-space I am sampling from? Is there a better way to sample $\vec{\mathbf{\gamma}}$ so that the values seem more uniformly distributed?
My linear algebra skills are not strong so this is proving to be very difficult. In the end, I need only to sample $\vec{\mathbf{\gamma}}$ such that $\sum_{i=1}^k\gamma_i=0$ and $\sum_{i=1}^k\gamma_i^2=\varphi^2$.