Does $v \cdot B$ carry physically meaning information? Given two vectors $v$ and $B$, $\frac{v \cdot B}{|B|}$ yields the magnitude of the projection of $v$ onto $B$ and $\frac{|v \times B|}{|B|}$ the magnitude of the rejection of $v$ from $B$.
The force experienced by a positive charge due to the magnetic field at some point is given by the Lorentz force law $F = v \times B$, where $v$ is velocity of the charge and $B$ is the magnetic field at that point.
This prompts me to wonder if $v \cdot B$ carries physically relevant information, as $v \times B$ does.
 A: It depends on what the velocity $\mathbf{v}$ refers to. In plasma physics and magnetohydrodynamics, if $\mathbf{v}$ is the fluid velocity, then the quantity $\mathbf{v}\cdot\mathbf{B}$ is called the "cross-helicity".  It turns out that it can impose constraints on the collective behavior of the plasma or conducting fluid.  For example, in astrophysical plasmas, large-scale magnetic fields are generated from turbulent motion in a process known as the "dynamo effect," and cross-helicity can reduce the rate at which the dynamo-generated field decays.
If $\mathbf{v}$ refers to a fluid of a single sign of charge, like electrons, then there is a current associated with it:  $\mathbf{J} = -n_e e \mathbf{v}$, where $n_e$ is the number density of electrons and $e$ is the electron charge.  In that case, the current produces a second magnetic field that circulates around the current, hence around the first field.  This is "magnetic helicity."  It is a measure of how knotted the magnetic field lines are.  this also imposes constraints on the dynamics.  In particular, if the fluid is ideal (resistivity is zero), then the total helicity of the system is a constant, so the dynamics are constrained by both energy conservation and helicity conservation.  That means the field lines cannot break or reconnect, so the topology of the field geometry cannot change, and that limits the magnetic configurations the system can achieve.
If $\mathbf{v}$ refers to a single particle, I am not aware of any meaningful physical interpretation. The particle is simply free to stream along the direction of the magnetic field.
