A cross-product (determinant) saved my life, or did it?

Your friend has been kidnapped by evil perfectionists, they put him in the middle of a square room, with two opposite walls painted one black one blue, and the other two painted white, there is a door at each white wall, one leads to instant death by vaporization and the other to freedom. In the middle of the ceiling there is a tiny circular hole from which your friend can receive your voice and send his and nothing else.
He is facing a black wall, and if he does not escape the room within one minute he will be vaporized. Everything that can be sensed by a human is perfectly symmetrical in that room. Your vulgar friend does not know his left from his right, but you know that the door to his right leads to freedom.
What is a better use of your time? Trying to lead your friend to the door, or going for a cup of coffee?

We are in the future, 10 years later. We have established contact with a large amount of different intelligent alien species, and you even befriended an alien. To celebrate the vaporization of your last friend, evil hopeless perfectionists kidnap your alien friend, and put him in the same situation. Everything that can be sensed by this alien is perfectly symmetrical in that room.
At this point I need to tell you a bit about him: his species is Platonicus Solidus, he is a perfect sphere. He has a circular bottom on which he can bounce or rotate, and a diametrically opposed top from which he can hear, and under which his perfectly spherical brain resides. His sight is handled by 4 perfectly spherical and identical eyes, arranged symmetrically along his equator, he can never tell them apart.
However, your friend can actually feel the difference between clockwise and counter-clockwise rotation! So the kidnappers think you have better chances.
You know that the right door is to your right if you were in there facing the black wall.
Is it time for another coffee?