Assuming that the sag of the new 120mm fork will be set to the same as the old 100mm fork and headset spacers will remain constant, will the geometry of my bike change moving from a 100mm travel fork to a 120mm of travel? Is this as easy as adding 20mm to the height of the front end, or does the "height" vary by make and model? Thanks.
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Since the headtube is at an angle, it will not raise the bike as much as if it was perpendicular to the ground. How much height that would add will depend on what angle your headtube is at, probably something around 70d.
Now, adding travel will cause your bike to have slower handling. It will make it more stable in a straight line, but more effort will be needed to change direction. It will also move you farther back in the cockpit with may make it lift the front wheel more on steep climbs.
Also, by adding this travel, it will probably void the warranty on your frame because it is putting stresses on the frame that are outside what it was designed for. There is also a possibility that you could stress the headtube to where it breaks, but 20mm probably wouldn't do it.
In practice, a 20mm travel gain is within acceptable use range for almost any bike. It will change the geometry, but only slightly.
It will also change how the bike handles for steering and braking, but again, only slightly.
If you aren't adding a huge amount of weight or leverage to the bike, there should be no issue with the frame.
You should make sure the setup is correct for the new fork. It will not have the same sag and spring rate(most likely), and should not.
You should probably take it a bit easy on your bike for a ride or two, while you get the feel for how your bike will handle with the new fork.
Aside from that, enjoy the new fork.