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Hello, I was wondering what that statement in the picture (underlined) means? I’m familiar with the term “order of magnitude”, i.e powers of 10 when a measured value is expressed in scientific notation. But what does the phrase in this case (in the picture) mean?

Similarly, in the question below, this phrase shows up again. Though there’s a difference, the question asks us to arrange the vectors ‘in’ order of their magnitude. From what I understand, all of those four vectors have the same magnitude, don’t they? Just their directions, i.e the angles they make with the three coordinate axes are different, from what I understan. Please correct me if I’m wrong. Thanks

Note : This has been taken straight out of (page no. 19) the book “university physics with modern physics 14th edition” by Young and Freedman.

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  • $\begingroup$ I would say that 4.00 is order of magnitude of 10^0, while 11 is o.o.m. 10^1, so you should order them taking this into account; the first vector for example may appear 10^0 but if you multiply by 2 it becomes 10^1 (6*2=12); the conclusion you underlined makes sense because summing few vectors of similar magnitude you can go up of one or two orders of magnitude, while subtracting you can go to the null vector, which in some sense is much smaller $\endgroup$
    – Francesco Bernardini
    Jan 5, 2019 at 16:53
  • $\begingroup$ Thanks for clearing that up. Now I understand what the underlined conclusion is trying to clarify. I still don’t understand the question below though. I mean, how am I supposed to arrange the four vectors in order of their magnitude? Can you explain it please? $\endgroup$
    – 4d_
    Jan 5, 2019 at 17:18
  • $\begingroup$ I may be drunk, but those vectors appear to me of exactly same length, so I’d say all of them are at first place; maybe it’s a trick question... $\endgroup$
    – Francesco Bernardini
    Jan 5, 2019 at 17:20
  • $\begingroup$ Exactly what I’m confused about. Thanks for the explanations :) $\endgroup$
    – 4d_
    Jan 5, 2019 at 17:22
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    $\begingroup$ A brilliant teacher of physics once told me these types of problems were designed not to teach physics, but to keep unqualified people out of med school. $\endgroup$
    – JEB
    Jan 5, 2019 at 17:36

1 Answer 1

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Both "order" and "magnitude" have multiple meanings and different ones are used here, so the two uses of the phrase "order of magnitude" on this page are completely different.

Two numbers are the same "order of magnitude" if they're roughly the same size, up to, say, around a factor of $10$ difference.

The magnitude of a vector is its length. To put a list of vectors in "order of magnitude", write down the shortest vector first, then the next shortest, and so on.

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  • $\begingroup$ Thanks, I think I get it now, but still I’m a bit confused. You said, “To put a list of vectors in "order of magnitude", write down the shortest vector first, then the next shortest, and so on.” The question below, don’t those four vectors have the same magnitude? How am I supposed to arrange them? $\endgroup$
    – 4d_
    Jan 5, 2019 at 17:14
  • $\begingroup$ Yes, @πtimese, it seems to be a trick question. Does the book have a section with the correct answers? $\endgroup$
    – md2perpe
    Jan 5, 2019 at 17:21
  • $\begingroup$ @πtimese Indeed, it's a trick question. This, combined with the fact that they used the same phrase "order of magnitude" twice in a page with two completely different meanings, makes me suspect that the authors of this textbook don't have the best interests of the students in mind. I certainly never ran into anything dumb like this when reading Halliday and Resnick, one of the competitors to Young and Freedman. $\endgroup$
    – knzhou
    Jan 5, 2019 at 17:23
  • $\begingroup$ It does not, or I’d have posted it on here in my original post. I even googled this question and although the website ‘Chegg’ has a solution for this question, the solution is locked and I don’t have a subscription. I couldn’t find this question anywhere else when I looked it up on google $\endgroup$
    – 4d_
    Jan 5, 2019 at 17:25
  • $\begingroup$ @knzhou thanks again, that might be the case. All four vectors do have the same length, i.e magnitude $\endgroup$
    – 4d_
    Jan 5, 2019 at 17:27

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