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I know that a linear array of microstrip patches can be either horizontal or vertical depending on its orientation.

Does the feed (input port of the array) determine the direction of polarization?

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The short answer is yes, the position of the feed of a single patch determines the horizontal vs. vertical polarization of that patch.

Since all the patches in the linear array are fed in the same way, this determines also the polarization of the array.

See this article, which covers the working of single-patch antennas:

Some excerpts:

Topics include principles of operation, impedance matching, radiation patterns, circular polarization, bandwidth, efficiency, alternative feed types, stacked patches and higher mode behavior.

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The relevant section for your question is the following:

Polarization

The plane in which the electric field varies is also known as the polarization plane. The basic patch covered so far is linearly polarized since the electric field varies in only one direction. This polarization can be anything between vertical and horizontal depending on the orientation of the patch. The polarization plane is the xz-plane in Figure 1. For optimum system performance, transmit and receive antennas must have the same polarization. The patch described above yields horizontal polarization and when rotated by 90°, the current flows in the vertical plane and the antenna is now vertically polarized.

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enter image description here

The previous image, which in the article is used to illustrate how to obtain a circular polarization, explains why the feed position determines the polarization: feed1 and feed2 excite different linearly polarized modes (TM10 and TM01), which combine to give circular polarization. From that it is clear (read the text in the article) that if only one feed is present, it excites just one linearly polarized mode.

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