What happens to relative humidity as the air temperature drops below the dew point?

Your question is ambiguous because there are two ways to look at it. If you take a parcel of air and increase its temperature, the relative humidity will decrease. If you take a parcel of air, the higher the temperature, the more moisture it can contain.

You see, a cubic meter of air at 0 C (freezing point) can't contain more than 5 grams water. When it happens, the relative humidity is 100 percent and the absolute humidity is 5 gr / m3. The same parcel of air at 15 C can contain 13 grams of water.

When it happens, the relative humidity is also 100 percent but the absolute one is 13 gr / m3. So, if you take a saturated parcel of air at 0 C and warm it up to 15 C, the relative humidity will drop 100 / 13 * 5 = 38 percent and that is relatively dry air. When the air reaches 100 percent saturation it is said to be at dew point temperature because this is when air can't sustain more water and if further cooled, must get rid of some of the water by condensing it into tiny droplets known as the dew.

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