Can We Conclude a=b from a.b=a.c?

[kidia]

help this:
If a is a given vector and a.b=a.c can we conclude that a=b?

 

[arildno]

Please read your exercise again.
I am quite certain you have written it down wrongly..

 

[dextercioby]

I'm sure it asks about

\vec{a}\cdot\vec{b}=\vec{a}\cdot\vec{c}\Rightarrow \vec{b} \ ? \ \vec{c}

Daniel.

 

[Data]

in which case, here's a hint: Let a = (1, 1). Can you think of two different vectors that when dotted with a give you 1?

 

[kidia]

I have written it correct, If a is a given vector and a.b=a.c can we conclude that a=b?

 

[chroot]

kidia,

In that case, it's a "trick question." You have no idea what c is, so you cannot conclude anything.

- Warren

 

[Data]

...

well, for any a, b you have a \cdot b = a \cdot b, and it's certainly possible to have a \neq b, so uhh... no.

 

[dextercioby]

BTW,the solution to the problem i proposed is

\vec{b}=\vec{c}+\vec{a}\times\vec{k},where \vec{k} is arbitrary...

Daniel.

 

[matt grime]

in 3 dimensions (i hate smilies but thought there shuold be some tongue in cheek indicator)

 

[kidia]

May be the answer is no we cannot conclude that a=b but I am not sure

 

[dextercioby]

On what basis could you draw a correct conclusion,given the problem in its form...?

Daniel.

 

[Antiphon]

The answer is no.

Let a=x-hat, b=y-hat, c=z-hat.

a.b = a.c = 0. Note that the 0 does not trivialize the result. You can
find infinite non-zero vectors with non-zero dot products that could
satisfy the relation.