π projects a 2D vector onto a line. It’s linear, so applied to the vector with coordinates x and y:
π(x e₁ + y e₂) = x π(e₁) + y π(e₂)
π(e₁) and π(e₂) are just numbers, so this is a dot product.
Takes more work to prove it’s a dot product with a unit vector on the line. And more assumptions about projection, beyond just linearity
im calling a map from 2D to 1D a “projection”, like from 2 numbers to 1 number. You may prefer to call that the length of a projection, idk
again, i only used linearity, none of the above is really specific to projection, but that’s what the drawing is of
