. . . trivial:¹

Strictly speaking, Eq. (10.69) is for a point charge, and so should be applied separately to each charge element $\lambda dz$. However, since $\Vec{v}$ has the same magnitude and direction for each, v comes outside the integral and all the non-y components of the individual cross products cancel out the same way the horizontal components of $\Vec{E}$ do.

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