What is the origin of the custom of opening the ark for important prayers (not connected with the need to take out the Torah scroll)?
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Levush Orach Chaim 133 says that this is done to help us have the proper intention by Tefila.
Rivivos Efraim Volume 3:395 says that the reason we open the Aron Kodesh has to do with the Posuk "Vataal Shaavosom El HoElokim ותעל שועתם אל האלקים" which is said regarding the place the Sefer Torah rests upon, and the proof is that we open it up for Shema Kolainu to inspire the Zechus of the Sefer Torah.