Mechanics 2: Linear and Rotational Dynamics - 3D Math Primer for Graphics and Game Development - page 615

12.5.3 3D Rotational Dynamics

Now let's extend the basic principles developed in Section 12.5.2 into three
dimensions. First, let's review the 3D rotational kinematics quantities.
The single angle θ is replaced by a rotation tensor of some kind, with a
rotation matrix R or a quaternion q being the most common methods
of describing orientation in general rigid body simulations. The angular
velocity ω and acceleration α become vector quantities and get bolded as
ω and α, respectively.
To extend the dynamics principles into three dimensions, we start with
torque. Not surprisingly, torque becomes a vector quantity denoted τ, and
the direction of this vector indicates the axis about which the torque is
tending to induce rotation. (Later we consider what happens if the object
is already rotating about a different axis.) The formula for computing the
torque for an applied force f and lever arm l is actually simpler in 3D than
the corresponding 2D formula!

Torque in Three Dimensions

τ = l × f . (12.29)

Compare Equation (12.29) to τ = Fl sinφ (Equation (12.24)), and notice
that the cross product has the magnitude and sinφ terms built in.
Angular momentum likewise becomes a vector L , with a similar formula
for its relation to the linear quantity:

Orbital angular
momentum of a particle
in three dimensions with
radial vector r

L = r × P .

Compare this to Equation (12.28).
A reader who is paying attention might note that Equation (12.28) is
only one of two equations we gave for angular momentum in the plane—
the one we deemed to be more appropriate for orbital angular velocity of a
particle—and wonder about the other formula, Equation (12.27), which was
more appropriate for spin angular velocity. That formula was L = Jω, and
to get its three-dimensional equivalent, we must understand how to extend
J, the moment of inertia, into three dimensions. Luckily, the link between
the two momentum equations is an excellent way to get this understanding.
Let's start by expanding L = r × P , with the goal of ending up with