Properties of cross and dot products

[bobey]

Homework Statement

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.

Homework Equations

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

 

[HallsofIvy]

Homework Statement

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.

Homework Equations

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

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.

 

[bobey]

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.

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 vectorIS MY ARGUMENT CORRECT NOW? or any other ideas...THANX 4 D RESPONSE HallsofIvy...

 

[Gavins]

Yes, that is right.