The discussion revolves around the proof of a vector equivalence involving the dot product, specifically examining the condition under which the equality a·b = a·c implies that b = c, given that vector a is non-zero.
The discussion is ongoing, with participants actively questioning the assumptions behind the vector equivalence. Some have provided specific examples to illustrate their points, while others are seeking clarity on how to express their reasoning effectively.
Participants note the importance of using specific vectors and dot products when constructing counterexamples, indicating a potential area of confusion regarding the generality of the statement being examined.
the method i have tried is really useless?radou said:Try to think of an example where it's not true.
assume b is not equal to cradou said:It's not useless, it just doesn't tell you anything.
You can solve this by finding a counterexample.
zero =[radou said:let a = (1, 0, 0), b = (1, 0, 1), c = (1, 2, 1).
What does a.(b - c) equal?
a.(b - c) = 0, when b doesn't equal cradou said: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.