# Multi-Variable Calculus: Cancellation of dot products

1. Sep 7, 2011

1. The problem statement, all variables and given/known data

In real-number multiplication, if uv1 = uv2 and u ≠ 0, then we can cancel the u and conclude that v1 = v2. Does the same rule hold for the dot product: If uv1 = uv2 and u ≠ 0, can you conclude that v1 = v2? Give reasons for your answer.

2. Relevant equations

3. The attempt at a solution

If we let u = k<u1, u2> with u2 = 0 and scalar k, then the dot product of u with any other vector v = k<v1, v2> will simply be the component kv1 because u2 will make the ku2kv2 product always zero regardless of its value. Thus, v can be infinitely many different vectors and still have the same dot product with u.

2. Sep 7, 2011

### Pyrrhus

The hint is can you compute an inverse of $\vec{u}$ ? and thus multiply both by that inverse in order that both vectors v1 and v2 are equal.

3. Sep 7, 2011

### lanedance

also a counter example might be good here, how about considering when
u • v1 = u • v2 = 0

what does this mean geometrically? using that it should be easy to find a counter example in 3D space

4. Sep 7, 2011

### vela

Staff Emeritus
Just out of curiosity, why do you introduce the scalar k? Doesn't your argument work without the k's?

5. Sep 7, 2011