# Given Vector

1. Apr 1, 2005

### kidia

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

2. Apr 1, 2005

### arildno

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

3. Apr 1, 2005

### dextercioby

I'm sure it asks about

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

Daniel.

4. Apr 1, 2005

### 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 $1$?

5. Apr 1, 2005

### kidia

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

6. Apr 1, 2005

### chroot

Staff Emeritus
kidia,

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

- Warren

7. Apr 1, 2005

### 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.

8. Apr 1, 2005

### 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.

9. Apr 1, 2005

### matt grime

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

Last edited: Apr 1, 2005
10. Apr 1, 2005

### kidia

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

11. Apr 1, 2005

### dextercioby

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

Daniel.

12. Apr 4, 2005

### 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.

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