## Problem on vector projections

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?
• ## Solution

#### Conceptual Proof

We begin by exploring the question conceptually.
Recall that
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 $\text{proj}_{c \bfv} \bfx = \text{proj}_\bfv \bfx$
Recall the formula for the projection:
We use this formula to also write $$\text{proj}_{c\bfv}\ \bfx = (\bfx\cdot c\bfv) \frac{c \bfv}{\left| c\bfv \right|^2}.$$
we can compute that \begin{align}\text{proj}_{c\bfv}\ \bfx &= (\bfx\cdot c\bfv) \frac{c \bfv}{\left| c\bfv \right|^2} = \frac{c^2}{|c|^2} (\bfx\cdot \bfv) \frac{\bfv}{|\bfv|^2}.\\
&= (\bfx\cdot \bfv) \frac{\bfv}{\left| \bfv \right|^2} = \text{proj}_{\bfv} \bfx
\end{align}
Note that $c^2 = |c|^2$ for any scalar $c$.
We conclude that the projection of $\bfx$ onto $\bfv$ does not depend on the length of $\bfv$.