Consider the following paragraph from p5 of this material
Now, choose a basis $w_1,w_2$ of $W$ so that $w_2$ is perpendicular to $v_1$. If $v_1$ is perpendicular to $W$, any unit vector in $W$ will do as $w_2$. If not, choose $w_2$ to be the unit vector in $W$ perpendicular to the projection of $v_1$ onto $W$.
It is saying the following points
- $w_2$ is perpendicular to $v_1$
- $w_2$ is perpendicular to the projection of $v_1$ onto $W$.
Are they both same? If yes, how can $w_2$ perpendicular to $v_1$ and to $v_1$'s projection on $W$?
They are not the same. Choose $v_1=(1,1,1)$ and consider $W$ to be the span of $w_1=(1,0,0)$ and $w_2=(0,1,0)$. Now the projection of $v_1$ onto the subspace spanned by $(1,0,0)$ is simply $(1,0,0)$ which is orthogonal to $w_2$ but $v_1 \cdot w_2 = 1 \neq 0$ so they are not orthogonal.