Is there meaning or application to taking the cross product of the first and the second derivative of a vector valued function? In my multivariable calculus class we have several practice problems where given a vector valued function r(t):
we need to find $$r'(t) \cdot r''(t)$$
and
$$r'(t) \times r''(t)$$
For example, using the given function 
$r(t) = cos(t)i +sin(t)j +2tk$
which is from Larson 10e section 12.2 example 2.
The book seems to emphasize this calculation in the exercises and because of the frequency with which it appears, I wonder if this calculation has any significance.  Is this just a convenient way for the authors to practice dot products, cross products, and derivatives of vector functions, or is there a meaning in math or any other field to taking the dot or cross product of a first derivative with a second derivative.  I know how to solve the calculation, my question is whether there is any meaning to it.
 A: The first of these quantities is proportional to the derivative of speed (squared):
$$
   \frac{d}{dt}|r'|^2 = \frac{d}{dt} r'\cdot r' = r'' \cdot r' + r' \cdot r'' = 2 r' \cdot r''
$$
I need to be a bit more hand-wavy about the second one.  Remember that
$
    |u \times v|
$
is the area of the parallelogram spanned by $u$ and $v$.  So $r'\times r''$ is a vector quantity whose length is proportional to $|r'|$, $|r''|$ and the sine of the angle between them.  If the angle between $r'$ and $r''$ is large (i.e., close to a right angle), there's a lot of curving to the curve.  
The curvature formula
$$
    \kappa = \frac{|r' \times r''|}{|r'|^3}
$$
comes from reparametrizing $r$ so that its speed is constant, and normalizing $r'$ to a unit tangent vector field $T= \frac{r'}{|r'|}$.  If $s$ is the arc length as a coordinate, then $\kappa = |T'(s)|$.  When you compute this derivative carefully, the $r'\times r''$ pops out, not for any mystical reason other than BAC-CAB.
A: In physics, angular momentum is defined as $L=m r(t) \times r'(t)$. Thus its derivative is
$$\frac {dL}{dt} =mr'(t) \times r'(t) + m r(t)\times r''(t)=mr(t) \times r''(t). $$
Thus the angular momentum is conserved, if $mr''(t)\times r(t)=r\times F=0.$
This happens for example if the force is a central force (gravitation of the sun, electric field of a single particle), i.e. $F(r)=|F| e_r$, where $e_r$ is the unit vector in radial direction. 
