Answer
Proof given below.
Work Step by Step
${\bf w_{1}}=\displaystyle \mathrm{p}\mathrm{r}\mathrm{o}\mathrm{j}_{{\bf v}}{\bf u}=(\frac{{\bf u}\cdot{\bf v}}{|{\bf v}|^{2}}){\bf v}.$
$\displaystyle \mathrm{p}\mathrm{r}\mathrm{o}\mathrm{j}_{{\bf v}}{\bf u}\cdot \mathrm{p}\mathrm{r}\mathrm{o}\mathrm{j}_{{\bf v}}{\bf u}= \frac{({\bf u}\cdot{\bf v})^{2}}{|{\bf v}|^{4}}\cdot|{\bf v}|^{2}= \frac{({\bf u}\cdot{\bf v})^{2}}{|{\bf v}|^{2}}.$
.
${\bf w_{2}}={\bf u}\displaystyle \cdot \mathrm{p}\mathrm{r}\mathrm{o}\mathrm{j}_{{\bf v}}{\bf u}=(\frac{{\bf u}\cdot{\bf v}}{|{\bf v}|^{2}}){\bf u}\cdot{\bf v}= \frac{({\bf u}\cdot{\bf v})^{2}}{|{\bf v}|^{2}}.$
$( {\bf u} -\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{j}_{{\bf v}}{\bf u})\cdot \mathrm{p}\mathrm{r}\mathrm{o}\mathrm{j}_{{\bf v}}{\bf u}={\bf w_{1}-w_{2}}$
$\displaystyle \frac{({\bf u}\cdot{\bf v})^{2}}{|{\bf v}|^{2}}-\frac{({\bf u}\cdot{\bf v})^{2}}{|{\bf v}|^{2}}=0$
