Show that if u dot v = u dot w for all u, then v = w (u, v, and w are vectors)?

Question

Preferably for any number of dimensions (R^n)

Answer 1

This is actually true in any Hilbert space. By subtracting we see that u*(v-w)=0 for all u. In particular, let u=v-w to find that 0=(v-w)*(v-w)=||v-w||^2. Thus, the distance between v and w is 0, so v=w.

Answer 2

Suppose ∀u, = . Then ∀u, = - = 0. So in particular, this holds for u=v-w -- i.e. = 0. This means that v-w=0, so v=w. Q.E.D.

Answer 3

Let u, v, and w be vectors in R^n. Hence,
u = u1, u2, u3, ..., un
v = v1, v2, v3, ..., vn
w = w1, w2, w3, ..., wn
From the definition of the direct product of two vectors,
u dot v = u1 * v1, u2 * v2, u3 * v3, ..., un * vn
u dot w = u1 * w1, u2 * w2, u3 * w3, ..., un * wn
Let u dot v = u dot w for all u; that is,
u1 * v1, u2 * v2, u3 * v3, ..., un * vn = u1 * w1, u2 * w2, u3 * w3, ..., un * wn
Hence,
u1 * v1 = u1 * w1, u2 * v2 = u2 * w2, u3 * v3 = u3 * w3, ..., un * vn = un * wn
By some law of multiplication or division, it must follow that
v1 = w1, v2 = w2, v3 = w3, ..., vn = wn.
Therefore, v = w.