- #1
kidia
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help this:
If a is a given vector and a.b=a.c can we conclude that a=b?
If a is a given vector and a.b=a.c can we conclude that a=b?
Can We Conclude a=b from a.b=a.c?
- Context: Undergrad
- Thread starter kidia
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Discussion Overview
The discussion revolves around the question of whether the equation \( \vec{a} \cdot \vec{b} = \vec{a} \cdot \vec{c} \) implies that \( \vec{b} = \vec{c} \) when \( \vec{a} \) is a given vector. The scope includes mathematical reasoning and conceptual clarification regarding vector dot products.
Discussion Character
- Debate/contested
- Mathematical reasoning
Main Points Raised
- Some participants question whether the original statement is correctly interpreted, suggesting it may be a trick question due to the unknown nature of \( \vec{c} \).
- One participant proposes a specific example with \( \vec{a} = (1, 1) \) and asks for two different vectors that yield the same dot product with \( \vec{a} \).
- Another participant asserts that it is possible to have \( \vec{a} \cdot \vec{b} = \vec{a} \cdot \vec{c} \) while \( \vec{b} \neq \vec{c} \), providing a counterexample using unit vectors in three dimensions.
- One participant expresses uncertainty, suggesting that the answer may be no, but is not entirely sure.
- Another participant emphasizes that without additional information about \( \vec{c} \), no conclusion can be drawn.
Areas of Agreement / Disagreement
Participants generally disagree on whether the conclusion \( \vec{b} = \vec{c} \) can be drawn from the given equation. Multiple competing views remain, with some asserting it is a trick question and others providing counterexamples.
Contextual Notes
Limitations include the lack of information about the vectors involved, particularly \( \vec{c} \), and the potential for multiple vectors to satisfy the dot product condition without being equal.
- #2
arildno
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Please read your exercise again.
I am quite certain you have written it down wrongly..
I am quite certain you have written it down wrongly..
- #3
dextercioby
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I'm sure it asks about
\vec{a}\cdot\vec{b}=\vec{a}\cdot\vec{c}\Rightarrow \vec{b} \ ? \ \vec{c}
Daniel.
\vec{a}\cdot\vec{b}=\vec{a}\cdot\vec{c}\Rightarrow \vec{b} \ ? \ \vec{c}
Daniel.
- #4
Data
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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
kidia
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I have written it correct, If a is a given vector and a.b=a.c can we conclude that a=b?
- #6
chroot
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kidia,
In that case, it's a "trick question." You have no idea what c is, so you cannot conclude anything.
- Warren
In that case, it's a "trick question." You have no idea what c is, so you cannot conclude anything.
- Warren
- #7
Data
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...
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.
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
dextercioby
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BTW,the solution to the problem i proposed is
\vec{b}=\vec{c}+\vec{a}\times\vec{k},where \vec{k} is arbitrary...
Daniel.
\vec{b}=\vec{c}+\vec{a}\times\vec{k},where \vec{k} is arbitrary...
Daniel.
- #9
matt grime
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in 3 dimensions 
(i hate smilies but thought there shuold be some tongue in cheek indicator)
(i hate smilies but thought there shuold be some tongue in cheek indicator)
Last edited:
- #10
kidia
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May be the answer is no we cannot conclude that a=b but I am not sure
- #11
dextercioby
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On what basis could you draw a correct conclusion,given the problem in its form...?
Daniel.
Daniel.
- #12
Antiphon
- 1,685
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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.
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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