Let the position vectors of three vertices of a triangle be 4 p + q -3 r,-5 p + q +2 r and 2 p - q +2 r. If the position vectors of the orthocenter…

Let the position vectors of three vertices of a triangle be $\displaystyle 4 \vec{p}+\vec{q}-3 \vec{r},-5 \vec{p}+\vec{q}+2 \vec{r}$ and $\displaystyle 2 \vec{p}-\vec{q}+2 \vec{r}$. If the position vectors of the orthocenter and the circumcenter of the triangle are $\displaystyle \frac{\vec{p}+\vec{q}+\vec{r}}{4}$ and $\displaystyle \alpha \vec{p}+\beta \vec{q}+\gamma \vec{r}$ respectively, then $\displaystyle \alpha+2 \beta+5 \gamma$ is equal to: