ABCDEF is a regular hexagon and O is the centre of the regular hexagon .
Vector OB = $b$ , Vector OC = $c$ . The side FA is produced to a point G such that FA : AG = 1 : 2.
The line DG meets OC at point X such that
vector DX = $h$ DG , where h is a constant . Express vector DX in terms of $h$, $b$, $c$,
Given also that vector OX = $k$ OC , where k is a constant , express vector OX in terms of $k$ and $c$,
hence find the value of $h$ and of $k$
I found : Vector DX = $2h ( -c + 2b ) $
Vector OX = $ kc$
Now I have 2 unknowns , I'm not sure on how to start to find unknowns $k$ and $h$, I thought of simultaneous equations but I don't know how to start. Can I get a hint ? Thanks in advance !
HINT: $$\vec{OD}=\vec{BC}=c-b$$ Now write $\vec{OX}$ as $\vec{OD}+\vec{DX}$, and then in terms of $b$ and $c$. For $\vec{OX}$ to be a multiple of $c$, the multiple of $b$ in its expression must be zero – what $h$ will achieve this? Once you've found it, substitute to find $k$.