Is it true that the slope of a vertical line times the slope of a horizontal like don't equal $-1$, even though they're perpendicular?

Question

I know that the slopes of two lines that are perpendicular have a value of $-1$ when multiplied because they're opposite reciprocals (e.g. $5$ and $-{1\over 5}$), but what if there's a horizontal and a vertical line ($x=3$ and $y=-2$). They're perpendicular, too, but a vertical line has no slope, so about $0\cdot\infty$, that would be undefined, right? Well, infinity is undefined, so that's why a vertical line has no slope. I also think this is why that silly problem is undefined and doesn't always equal zero. Well, what's the slope of a vertical line times the slope of a horizontal line? Tell me what you think!