resolve to perpendicular components, because they are independent

Hi Physicers,
Any help will be greatly appreciated.
It is my understanding that we often resolve vectors (like force,
velocity) into perpendicular components because they are so called
"independent." If one component changes value, it doesn't affect the
value of the other pependicular components (I guess this is where
"Independent comes from").
Yet I have been recently shown how to resolve vectors (force) into
components that aren't perpendicular using a reverse-parallelogram
rule. Say a force is acting on a structure like < at the left point, I
have to resolve it into two forces alone the two branches (not
orthogonal). I encountered this in my self study of certain questions,
so I don't really have anyone reliable to ask why this is viable in
light of my previous knowledge of "independent vectors need to be
perpendicular."
Can someone shed light on any of this?

.

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