Question

Suppose we know that u · v = u · w. Does it follow that v = w? If it does, give a proof that is valid otherwise, give a counterexample (i.e., a specific set of vectors u, v, and w for which u · v = u · w but v is not equal to w)

Answer 1

No, the equality of dot products u · v = u · w does not imply that vectors v and w are equal. A counterexample with u = (1, 0), v = (0, 1), and w = (0, -1) shows that they can have the same dot product with u while being different vectors.

Step-by-step explanation:

The question is whether u · v = u · w implies that v = w. In vector algebra, two vectors being equal means their corresponding components must be equal. However, just because the dot products are equal does not necessarily mean the vectors themselves are equal, because the dot product only provides information about the projection of one vector onto another. For example, consider u = (1, 0), v = (0, 1), and w = (0, -1). In this case, u · v = u · w = 0, but clearly, v
eq w. Hence, this is a counterexample demonstrating that u · v = u · w does not necessarily imply that v = w.