Magnitudes of the sum of two vectors

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Discussion Overview

The discussion revolves around the validity of a statement from a textbook regarding the magnitudes of the sums of two vectors. Specifically, it questions whether the equality of the magnitudes of the sums of two vectors implies that the magnitudes of the individual vectors are equal. The scope includes conceptual reasoning and potential errors in mathematical reasoning.

Discussion Character

  • Debate/contested

Main Points Raised

  • One participant questions the textbook's assertion that if |a+b| = |a+c|, then |b| must equal |c|, suggesting that this is false based on their visual representation.
  • Another participant provides specific vector examples (a=(2,0), b=(1,0), c=(-5,0)) to illustrate their point, although the relevance of these examples is not fully explored.
  • Several participants express skepticism about the textbook's reasoning, particularly the claim that both scenarios represent the diagonals of parallelograms with equal sides.
  • One participant emphasizes that the statement is not even true in one dimension, reinforcing their disagreement with the textbook.
  • A later reply acknowledges a potential misunderstanding regarding the nature of the question in the textbook, suggesting it was misinterpreted as a statement rather than a question.

Areas of Agreement / Disagreement

Participants generally disagree with the textbook's assertion, with multiple viewpoints presented regarding the validity of the claim. The discussion remains unresolved as participants explore different interpretations and examples.

Contextual Notes

There is a lack of consensus on the interpretation of the textbook's statement, and the discussion highlights potential misunderstandings and the need for clarity in definitions and assumptions regarding vector magnitudes.

keroberous
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This is a question that I saw in a textbook:

"If the magnitude of a+b equals the magnitude of a+c then this implies that the magnitudes of b and c are equal. Is this true or false?"

The textbook says that this statement is true, but I'm inclined to believe it is false. I made a quick sketch to show my thinking visually.
PXL_20210610_180527814.jpg

I drew these diagrams to scale, so vector a is the same in each case and the lengths of a+b and a+c are in fact equal (both 5 cm). It's clear to me that b and c are different lengths/magnitudes here. I'm not sure if the text made an error (not unheard of) or if I made an incorrect assumption somewhere. Thanks!
 
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20210610_113118~2[1].jpg
 
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Suppose a=(2,0), b=(1,0), c=(-5,0) ...
 
So your diagram isn't all that different than mine, so I take it then that the textbook is incorrect and the statement is false?

Here's the book's entire reasoning:

"true; |a+b| and |a+c| both represent the lengths of the diagonal of a parallelogram, the first with sides a and b and the second with sides a and c; since both parallelograms have a as a side and diagonals of equal length |b|=|c|"
 
keroberous said:
So your diagram isn't all that different than mine, so I take it then that the textbook is incorrect and the statement is false?

Here's the book's entire reasoning:

"true; |a+b| and |a+c| both represent the lengths of the diagonal of a parallelogram, the first with sides a and b and the second with sides a and c; since both parallelograms have a as a side and diagonals of equal length |b|=|c|"
It's hard to think of anything more wrong!

It's not even true in one dimension!
 
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Maybe you should get different textbook?

Edit: Oops. I just noticed this was a question in the book, not a statement. It's just a typo. So - never mind...
 
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PeroK said:
It's hard to think of anything more wrong!

It's not even true in one dimension!
I'm glad I wasn't going crazy!

DaveE said:
Maybe you should get different textbook?
If only that was an option. lol

Thanks!
 
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