I'm wondering whether the residual light of the Big Bang comes from one particular direction and what possibilities do we have to detect its position?
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By "the light from the big bang", you must mean the cosmic microwave background, which did not come from the big bang directly, but was emitted during recombination about 400,000 years after the big bang. At that time, it was emitted from basically all directions and locations in space. The universe was born hot, and cooled gradually as it expanded, meaning the photons were becoming gradually less energetic. Photons and matter in the universe were interacting constantly, and light could not travel very far before interacting with protons and electrons and changing direction. Electrons and protons were also coming together to form neutral hydrogen atoms, but these were quickly dissociated by photons.
The cosmic microwave background was emitted when there were no longer enough photons with sufficient energy to break neutral hydrogen apart in to free protons and electrons. Once this became true, photons just kept on traveling in whatever direction they were going, without interacting. As this happened everywhere in the universe at just about the same time, you can see light that is part of the cosmic microwave background from any point in the universe, that will happen to show you light that is 13.7 billion years old (the time in the past when the cosmic microwave background was emitted).