Scalar product with ON-base $e_1,e_2,e_3$ Get the vector u which length is 4, in the ON-base $$e_1,e_2,e_3$$ and the baseangles $$\frac{\pi}{3}, \frac{5\pi}{6}, \frac{\pi}{2}$$

I assume you with baseangles mean the angle between $u$ and each basis vector $e_i$? 
If so, then this is just a matter of solving a set of  equations with three unknowns.
We known that $v \cdot w = |v| |w| \cos(\varphi)$ where $\varphi$ is angle between $v$ and $w$. If we let $u=(x,y,z)$, then using the fact just mentioned, we get three equations:
$$
\begin{array}
\, u \cdot e_1 & = & 4\cos \big(\frac{\pi}{3}\big) \\
\, u \cdot e_2 &=& 4\cos\big(\frac{5 \pi}{6}\big) \\
\, u \cdot e_3 &=& 4 \cos\big(\frac{\pi}{2}\big)
\end{array}
$$
I'll let you do the rest (which is not much).
