Slope of a line passing through $\left(7,17\right)$ and $\left(15,\beta \right)$ is $\frac{17-\beta }{7-15}$.

Since, both lines are perpendicular to each other, so product of slopes will be $-1$.

$\Rightarrow \frac{17-\beta }{7-15}\times \frac{2}{3}=-1$

$\Rightarrow \frac{17-\beta }{-8}=-\frac{3}{2}\Rightarrow 17-\beta =12\Rightarrow \beta =5$.