How many times are the hands of a clock at right angles in a day?

The minute hand moves at 6 degrees per minute and the hour hand at 0.5 degrees per minute, so the minute hand gains on the hour hand at a constant relative rate of 5.5 degrees per minute. Within any relative revolution of 360 degrees, the two hands are exactly at right angles, 90 degrees apart, precisely twice: once as the minute hand approaches from behind and once as it pulls further ahead.

1. Relative speed of the minute hand over the hour hand = 6 minus 0.5 = 5.5 degrees per minute.

2. Time for one complete 360-degree relative revolution = 360 divided by 5.5 = 720/11 minutes.

3. Number of complete relative revolutions in 12 hours (720 minutes) = 720 divided by (720/11) = 11.

4. Each relative revolution contains exactly two right-angle instants, so 12 hours contain 11 times 2 = 22 right-angle instants.

5. A full day has two twelve-hour halves, so the total for 24 hours is 22 times 2 = 44 right-angle instants.

This matches the related clock fact that the hands coincide only 11 times, not 12, in 12 hours, since the final coincidence of one half exactly meets the first coincidence of the next half; the same one-less-than-naive pattern governs right angles, confirming 22 per twelve-hour half and 44 across a full day.