no. because v and w are orthogonal, therefore you can use pythagoreans theorem to show ||u||^2 = ||w||^2 + |<u,v>/<v,v>|^2 · ||v||^2

however i do see a mistake on my part which is
|<u,v>|^2 = ||w||^2 + |<u,v>/<v,v>|^2 · ||v||^2 ||v||^2
|<u,v>|^2 = ||w||^2 + ||u||^2 ||v||^2

i forgot to distribute the ||v||^2 to the ||w||^2 which makes it incorrect

you are supposed to divide by ||v||^2 on both sides.

|<u,v>|^2 / ||v||^2 = ||w||^2 + |<u,v>/<v,v>|

and this shows that ||w||^2 = 0. therefore w=0