Magnitudes of the sum of two vectors

In summary, the textbook claims that if the magnitude of a+b equals the magnitude of a+c, then the magnitudes of b and c must also be equal. However, after creating visual diagrams and providing a counterexample, it is clear that this statement is false. The textbook's reasoning, which involves parallelograms, is also incorrect. It may be beneficial to seek a different textbook for more accurate information.
  • #1
keroberous
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1
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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  • #2
20210610_113118~2[1].jpg
 
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  • #3
Suppose a=(2,0), b=(1,0), c=(-5,0) ...
 
  • #4
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|"
 
  • #5
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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  • #6
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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  • #7
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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1. What is the formula for finding the magnitude of the sum of two vectors?

The magnitude of the sum of two vectors is equal to the square root of the sum of the squares of the individual magnitudes plus twice the product of the magnitudes and the cosine of the angle between them.

2. How do you calculate the magnitude of the sum of two vectors?

To calculate the magnitude of the sum of two vectors, you will need to first find the individual magnitudes of the vectors. Then, use the formula: √(A² + B² + 2ABcosθ), where A and B are the individual magnitudes and θ is the angle between them.

3. Can the magnitude of the sum of two vectors ever be negative?

No, the magnitude of the sum of two vectors can never be negative. Magnitude is a measure of the size or length of a vector, and it is always a positive value.

4. How does the direction of the vectors affect the magnitude of their sum?

The direction of the vectors does not affect the magnitude of their sum. The magnitude is only influenced by the individual magnitudes and the angle between them.

5. Is there a shortcut method for finding the magnitude of the sum of two vectors?

Yes, there is a shortcut method known as the parallelogram method. This involves drawing a parallelogram using the two vectors as adjacent sides, and the diagonal of the parallelogram represents the sum of the two vectors. The magnitude of this diagonal can be calculated using the Pythagorean theorem.

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