Why does a rotating wheel swerves to a certain direction when tilted? Why is right-hand rule for angular momentum true? 
My question is why does the wheel swerve to the right but not any other direction. Up to this point of the book, I've always thought the right-hand rule to find the direction of angular momentum is just a convention with no practical use. This conceptual problem blew my mind. Another way to express my confusion is "why is delta L's direction along the x-axis based on the right-hand rule", or what's the logical justification for the right-hand rule. Thank you very much! First time posting a question here. This website is gonna be a great help in my journey toward getting an engineering degree.
 A: The vector $\Delta\bf L$ is not one which if you point your thumb along it, then this means rotation of the wheel moves around your thumb in accordance with the right hand rule. It is simply a vector which indicates that the angular momentum direction, and therefore axis of rotation, has changed.
You're right that the angular momentum vector is a pseudo-vector. It is the the cross product of a displacement vector and a momentum vector, so that the result is a pseudo-vector. The direction of $\bf L+\Delta L$ represents the displacement of the original vector $\bf L$ to its new orientation due to moving it in the direction of $\Delta\bf L$.
In this sense, all $\Delta\bf L$ represents is the moving of the original axial vector $\bf L$. So imagine using the right hand rule with your thumb initially pointing along $\bf L$ and then move your thumb along the direction of $\Delta\bf L$ noting that the whole time your fingers still rap around the axis of rotation, until your thumb arrives to the new orientation given by $\bf L+\Delta L$.
Pointing your actual thumb in the direction of $\Delta\bf L$ and thinking this somehow means that the wheel is rotating around it clearly can't be right. But your question is justified, and is a good one. Remember the concepts of axial (pseudo) vectors as not being "entirely physical" as one would think the way you have, when first encountering such things.
A: The explanation shown in the text is incomplete. Indeed, the new angular momentum after the small torque is applied will be in the direction shown in the image, and it “points to the right, along the axle of the wheel.” Yet the answer implicitly assumed that the angular momentum must be directed only along the axle of the wheel.
The naive guess is that the angular momentum would in fact be still primarily “pointing to the right” as shown in the image, but that it will now have two “vector components,” one along the axle of the wheel and pointing only in the y direction, another causing it to “rotate up,” pointing in the x direction.
In other words, we imagine that it is possible for the wheel to spin along its axle and also rotate in the y-z plane, but this turns out not to be a stable configuration for the wheel to rotate in.
Some more food for thought: imagine instead of a wheel we had a long cylinder in zero-g, with its axis along the y axis. If the cylinder weren’t spinning initially and we applied forces at the ends causing the wheel to rotate in either the y-z plane or x-y plane, then the cylinder would keep rotating in that direction indefinitely. But when the cylinder is spinning, you can see that if you only change the angular momentum by a given amount and then cease making any new changes, the cylinder can’t keep spinning “about its axis” as well as have some other rotation overlapped on top of this, because that would imply the angular momentum is continually changing.
