Proof of Vector Equivalence: a‧b=a‧c

[SOHAWONG]

1.Determine if it is true that for any vectors a, b, c such that
a is not equal to 0 and a‧b = a‧ c, then b = c.



i tried to let a‧b-a‧c=0
then a‧(b-c)=0
but i found it's not meaningful
so how can i solve it =[
thz

 

[radou]

Try to think of an example where it's not true.

 

[SOHAWONG]

Try to think of an example where it's not true.

the method i have tried is really useless?

 

[radou]

It's not useless, it just doesn't tell you anything.

You can solve this by finding a counterexample.

 

[SOHAWONG]

It's not useless, it just doesn't tell you anything.

You can solve this by finding a counterexample.

assume b is not equal to c
l.h.s=a‧b
=...
should i express the dot product?

 

[radou]

let a = (1, 0, 0), b = (1, 0, 1), c = (1, 2, 1).

What does a.(b - c) equal?

 

[SOHAWONG]

let a = (1, 0, 0), b = (1, 0, 1), c = (1, 2, 1).

What does a.(b - c) equal?

zero =[
btw,when we want some example for conradiction,we should use some real number to think about it first?

 

[radou]

It doesn't matter what you use.

If you're not talking about a specific set of vectors and a specific dot product, and if you assume that, for any non-zero a, and any b, c, the implication a.(b - c) = 0 ==> b = c holds (which is equivalent to a.b = a.c ==> b = c) then it doesn't matter which vector space and dot product you chose to construct your counterexample.

So, we found an example where a.(b - c) = 0, when b doesn't equal c.

 

[SOHAWONG]

It doesn't matter what you use.

If you're not talking about a specific set of vectors and a specific dot product, and if you assume that, for any non-zero a, and any b, c, the implication a.(b - c) = 0 ==> b = c holds (which is equivalent to a.b = a.c ==> b = c) then it doesn't matter which vector space and dot product you chose to construct your counterexample.

So, we found an example where a.(b - c) = 0, when b doesn't equal c.

a.(b - c) = 0, when b doesn't equal c
this i'd thought once,but don't know how to express
anyway thank you very much :p