Vectors $\vec{b}$ and $\vec{c}$ are given. ${\sphericalangle} (b,c)=2\pi/3.$
Find vector $\vec{a}$, coplanar with $\vec{b}$ and $\vec{c}$, length $|\vec{a}|=4$ and ${\sphericalangle}(a,b)=\pi/6$.
I know it's something with triple product. Not sure where that gets me.
EDIT: $|\vec{b}|$=$|\vec{c}|$=1
If the angle between $a$ and $c$ is taken to be $\frac{\pi}{2}$, first find $n=b\times c$ to get the normal to the plane, and then find $c\times n$ to get the direction of $a$. Normalize $a$ and multiply by 4 and that's it.
If the angle between $a$ and $c$ is taken to be $\frac{\pi}{6}$, then $a$ is the angle bisector of $b$ and $-c$, so $a=\frac 12\left(\frac{a}{|a|}-\frac{c}{|c|}\right)$