If $u$ and $v$ are vectors orthogonal to $q$ and $c$ is a scalar, then $q ·(u − cv) = 0$
I created the diagram below to understand what's going on:
Questions:
When I see c, am I correct in assuming no matter what value it takes - ie no matter how each vector is scaled by c - the dot proudct will always be zero? (I know that even if c=0, the dot product will be zero.
Is drawing a picture the best way to look at this or is there a better way to deal with these sorts of conceptual questions?
Yes. No matter the value of $c$, the dot product will be zero. To prove it, you can distribute the dot product: $$q\cdot(u-cv) = q\cdot u -q\cdot(cv) = q\cdot u - c(q \cdot v) = 0.$$ Both dot products on the RHS are zero b/c q is orthogonal to both u and v.
Conceptually, the sum $u-cv$ will always be along the same direction as $u,v$.