Suppose that a regular tetrahedron with vertices $A$, $B$, $C$, and $D$ has its centroid at the origin $O$, as in the below schematic. Vectors $OA$, $OB$, $OC$, and $OD$ each have length $\ell$ ($|OA| = |OB| = |OC| = |OD| = \ell$). I wish to determine the four Cartesian vectors $OA$, $OB$, $OC$, and $OD$, given the axes shown in the figure. How can I do this?
Vector $OA$ is clearly $OA = (0, 0, \ell)$.
Vector $OB$ has no $x$ component. From chemistry (see, for example, this Wikipedia article), I know that the vectors make angles of $\cos^{-1}(-1/3) \approx 109.471^{\circ}$ with respect to one another. How should I determine $OB$, $OC$, and $OD$? Thanks for your time.