Does the vector projection of $\bfx$ along $\bfv$ depend on the length of $\bfv$? That is, if $\bfv$ is scaled by a scalar $c$, does the projection change?
If we scale the length $\bfv$, the point obtained by dropping down a perpendicular line from $\bfx$ is unchanged. Hence the projection of $\bfx$ onto $\bfv$ is does not depend on the length of $\bfv$.
Algebraic Proof
We would like to show that the projection of $\bfx$ onto $\bfv$ is the same as the projection of $\bfx$ onto the scalar multiple $c \bfv$.
That is, we want to show that $\text{proj}_{c \bfv} \bfx = \text{proj}_\bfv \bfx$
