= Solution
Unpolarized incident light has equal, mutually incoherent $y$- and $z$-polarized intensities. Both components scatter into the same direction $\mathbf v$, so their <coherency matrix>[coherency matrices] add to
$$
\mathbf J_s=\frac{KI_0}{2}(u_y^2+u_z^2)\begin{pmatrix}u_x^2&u_xu_z\\u_xu_z&u_z^2\end{pmatrix}.
$$
This matrix has rank one when the signal is nonzero. The light is therefore \b[fully linearly polarized along $\mathbf v$], with
$$
\boxed{I_{\rm unpol}=\frac{I_a+I_b}{2}=I_R.}
$$
A fixed highly anisotropic needle can act as a polarizing scatterer even for unpolarized input. The equality of total intensities with (c) does not mean the incident states were identical. These conclusions use the leading longitudinal-polarizability approximation; finite transverse response can change the <polarization> purity.
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