vector question.?

Question

suppose vector u dot v= w dot v for all v. Is it necessarily true that u=w? justify ur answer.

note: u,w,v are vectors.
dot= dot product

Answer 1

u dot v = |u| |v| cos (theta)
w dot v = |w| |v| cos (theta)

It is not necessarily true that u = w
the dot products may be equal, but u and w may have different magnitudes and angles with v

the dot product is only a scalar representation of how similar or dissimilar the directions of two vectors are

Answer 2

No. For instance all perpendicular vectors to v have dot products of 0, and there are a lot.