Consider an arbitrary coordinate system (not necessarily orthogonal) with origin $O$ and a triangle $AB\Gamma$. Let $\gamma$ and $\alpha$ be points in the sides $AB$ and $B\Gamma$ respectively such that $$\vec{O\gamma} = \frac{1}{5}\vec{OB} + \frac{4}{5}\vec{OA} \quad\text{ and } \quad \vec{O\alpha}=\frac{1}{3}\vec{O\Gamma} + \frac{2}{3}\vec{OB}$$ Let $\beta$ be the point where the lines $A\Gamma$ and $\alpha\gamma$ intersect. Find $\kappa, \lambda \in \mathbb{R}$ such that $\vec{O\gamma} = \kappa \vec{OA}+\lambda \vec{O\Gamma}$
Attempt: Using Menelaus's theorem I was able to deduce that $\vec{O\beta}=-\frac{1}{7}\vec{O\Gamma}+\frac{8}{7}\vec{OA}$. From there I tried playing with the three relations but nothing fruitful came up. Any help is appreciated. I am also attaching the figure below for clarity
