Current-carrying wire in a magnetic field. Cross product, vectors and scalars

We have a wire with cross-sectional area $$A$$, length $$L$$ and current $$I$$.

If the wire is in a magnetic field $$\vec B$$, the magnetic force on each charge is $$\vec F =q\vec v_d \times \vec B$$.

$$\vec v_d$$ is the drift velocity or average velocity

The number of charges in the wire segment is the number $$n$$ per unit volume, multiplied by the volume $$AL$$.

Thus the total force in the wire segments is $$(1)\: \:\vec F =(q\vec v_d \times \vec B)nAL$$

We know that the current in the wire is $$(2)$$ $$I=nqv_d A$$

Combining (2) and (1) the force can be written as $$(3)$$ $$\vec F=I\vec L \times \vec B$$

This is a paragraph from my textbook.

I know that the cross product has the property $$(rA \times B)=(A \times rB) = r(A \times B)$$

What I don't understand is why $$\vec v_d$$, that is a vector in (1) becomes a scalar in (3) as a part of $$I$$. And why, on the other hand, $$L$$ that is an scalar in (1) becomes a vector in (3). What property allows you to do that?

• (3) should probably be something like $\vec{F}=I\hat{\ell}\times\vec{B}$, where $\hat{\ell}$ is a unit vector in the direction of the axis of the wire. The reason being that $I$ is a scalar, but you want to have a vector pointing in the direction the current is flowing. – The Photon Aug 22 at 22:37