I think there might be something wrong with part (i) of your question.
Let's assume (taking a very specific case) in 2-D Cartesian:
vec u = (0,2)^T→u=(0,2)T, vec v = (0,1)^T→v=(0,1)T and vec a = (1,0)^T→a=(1,0)T
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Assuming, in the usual course, that your notation means that:
Proj_a vec u = Proj_a vec v implies vec u cdot hat a = vec v cdot hat aProja→u=Proja→v⇒→u⋅ˆa=→v⋅ˆa
...where hat aˆa is the unit vec a→a vector , then we have:
vec u cdot hat a = (0,2)((1),(0)) = 0
vec v cdot hat a = (0,1) ((1),(0)) = 0
So Proj_a vec u = Proj_a vec v = 0, which is your requirement in this very specific case.
But:
vec u cdot vec v = (0,2)((0),(1)) = 2
And:
vec v cdot vec a = (0,1)((1),(0)) = 0
So:
vec u cdot vec v ne vec v cdot vec a
The second bit (ie: part(ii)) is actually a lot easier to show once you add on the necessary conditions.
