Up is the opposite of down. My friend Juhani asked, why in the mirror the left becomes right but up doesn't become down. Fortunately I was graguated from Maths, I have no idea though...
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Here I assumed the mirror is on front of you. More generally, the mirror reverses the directions toward and away from the mirror. If the mirror is on the ceiling, it reverses up and down. Left and right are defined with respect to both back/front and up/down.
In terms of 3D vector algebra, left is the vector product (cross product) of up and forward: left = up x forward, where left, up, and forward are unit vectors pointing in the named directions.
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The massive musical organs in churches, how did they work 800 years ago?