I'm taking the unusual step of posting an additional answer, just to explain why a particular widespread explanation falls short.
Here is how I understand this type of attempted explanation.
To establish a pattern:
Uniform circular motion due to a centripetal force:
With uniform circular motion the acceleration vector is perpendicular to the momentum vector. The momentum vector turns at a constant rate in the direction of the acceleration vector, and thus uniform circular motion is sustained.
The counterpart of that in angular mechanics is depicted in the diagram that was supplied by Aaron Stevens.
A torque vector (green arrow) represents a particular force pair. Here there is the downward force of gravity (red arrow) and there is the upwards force from the triangular block that supports the gyroscope. The combined effect is represented with the torque vector.
In the diagram the torque vector and the angular momentum vector are perpendicular to each other. The suggested precession law is then as follows: the angular momentum vector turns at a constant rate in the direction of the torque vector, and thus gyroscopic precession is sustained.
I will now discuss why this way of representing gyroscopic precession is flawed.
First let me recapitulate the actual behavior of a gyroscope in a range of circumstances.
A If the gyroscope wheel is not spinning then when a force is applied the gyroscope will simply fall in the direction of the force.
B If the gyroscope wheel is spinning slowly then when a force is applied the gyroscope will fall the same as in case A, the difference will be negligable.
C An intermediate spin rate such that you do get a noticable difference with case A, but not enough to look like gyroscopic precession.
D When the gyroscope wheel is spinning sufficiently fast you get that striking effect that the gyroscope swivels rather than pitching down.
To anyone reading this: I very much encourage you to obtain a gimbal mounted gyroscope and start trying out the above range of circumstances.
The range from A to D stretches across a transition. An explanation of gyroscopic precession must explain the behavior for all of the range from A to D in one unified representation. You cannot dismiss cases A, B and C. An attempted explanation that seems to work for case D, but that fails for A, B and C is inadmissible.
With the behavior recapitulated I return to the suggested precession law:
"The angular momentum vector turns at a constant rate in the direction of the torque vector, and thus gyroscopic precession is sustained."
How should the suggested precession law be applied in case B and C?
Case B: the gyroscope wheel is spinning slowly, so the angular momentum vector is non-zero. Is the suggested precession law only valid above some particular spin rate, or is it valid for every spin rate, no matter how small.
If the suggested precession law is only valid above some particular spin rate, then what is the explanation of that?
There is another aspect:
Let's look at the following hypothetical case:
Let's assume that suggested precession law is as valid as Newton's laws of motion are.
That would imply that the start of gyroscopic precession is instantaneous. It would imply that truly the gyroscope wheel starts swiveling instead of doing any pitching.
That raises the question: where does the angular momentum of the precessing motion come from? The torque (green arrow) is perpendicular to the angular momentum vector of the precessing motion, so the angular momentum of the precessing motion cannot come directly from the torque (green arrow).
In the Feyman lectures on physics Feynman discusses how exactly gyroscopic precession starts, and he points out that the transition to precessing motion does not happen instantaneously. It cannot happen instantaneously, that would violate the laws of motion.
Feynman lectures, chapter 20 Rotation in space
Some people like to say that when one exerts a torque on a gyroscope,
it turns and it precesses, and that the torque produces the
When the motion settles down, the axis of the gyro
is a little bit lower than it was at the start. Why? (These are the
more complicated details, but we bring them in because we do not want
the reader to get the idea that the gyroscope is an absolute miracle.
It is a wonderful thing, but it is not a miracle.) If we were holding
the axis absolutely horizontally, and suddenly let go, then the simple
precession equation would tell us that it precesses, that it goes
around in a horizontal plane. But that is impossible! Although we
neglected it before, it is true that the wheel has some moment of
inertia about the precession axis, and if it is moving about that
axis, even slowly, it has a weak angular momentum about the axis.
Where did it come from? If the pivots are perfect, there is no torque
about the vertical axis. How then does it get to precess if there is
no change in the angular momentum? The answer is that the cycloidal
motion of the end of the axis damps down to the average, steady motion
of the center of the equivalent rolling circle. That is, it settles
down a little bit low. Because it is low, the spin angular momentum
now has a small vertical component, which is exactly what is needed
for the precession. So you see it has to go down a little, in order to
go around. It has to yield a little bit to the gravity; by turning its
axis down a little bit, it maintains the rotation about the vertical
axis. That, then, is the way a gyroscope works.
Svilen Kostov and Daniel Hammer have done a table-top experiment to verify Feyman's assertions. The article describing their results is named after the key observation in Feynman's discussion 'It has to go down a little, in order to go around'