The line joining (1, 3) and (4, k) is perpendicular to the line $2x-3y+7=0$. What is the value of k?

Writing $2x-3y+7=0$ as $y=(2x+7)/3$ shows its slope is 2/3. A perpendicular line has slope −3/2. So $(k-3)/(4-1)=-3/2$, giving $k-3=-4.5$ and $k=-1.5$. The value $k=5$ is what you get if you make the lines parallel instead of perpendicular.