# Properties of cross and dot products

1. Sep 20, 2010

### bobey

1. The problem statement, all variables and given/known data
let u be a nonzero vector in space and let v and w be any two vectors in space. if u.v = u.w and u x v = u x w, can you conclude that v=w? give reason for your answer.

2. Relevant equations

3. The attempt at a solution
v is not necessary equal to w
since u is nonzero vector thus the cross product of the vector u with v and w will produce another vector and it is consider the same theirs magnitude and directions are same and their dot product will produce a scalar which has no indication of direction. thus v and w is not necessary same vector.

is my argument is true... or can anyone improve it and prove it with example... your help is highly appreciated.... thanx

2. Sep 20, 2010

### HallsofIvy

I have no clue what that means. You start by saying v is not necessarily the same as w, say a couple of general things about cross product and dot product and conclude that v and w are not necessarily the same. What, in what you said about cross product and dot product, led to that conclusion? When you say " it is consider the same their magnitude and directions are same" what vectors are you talking about having the same magnitude and direction? What does "their" refer to?
I think you would be better off using specific formulas. In particular I would recommend using the fact that $u\cdot v= |u||v|cos(\theta)$ and $|u\times v|= |u||v| sin(\theta)$.

It will help to know that for $\theta$ and $\phi$ both between 0 and 180 degrees, $tan(\theta)= tan(\phi)$ implies $\theta= \phi$.

3. Sep 21, 2010

### bobey

why you think to use this fact : $u\cdot v= |u||v|cos(\theta)$ and $|u\times v|= |u||v| sin(\theta)$ in your arguments??? I CAN'T SEE IT!!!

this is my new argument ::

yes v = w.
u.v = v.u and u x v = u x w implies that u.(v-w)=0 and u x (v-w) = 0 implies that u perpendicular with (v-w) and u parallel to (v-w) implies that u = 0 or (v-w) = 0

thus v=w since u is not a zero vector

IS MY ARGUMENT CORRECT NOW???? or any other ideas...

THANX 4 D RESPONSE HallsofIvy... :tongue:

4. Sep 21, 2010

### Gavins

Yes, that is right.