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Can we add any two vectors? If not, why is that so?

I think this is not true, but I am not sure. My book says it is true, but I guess it is a misprint. For example, adding acceleration to velocity.

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    $\begingroup$ Can you add 3 kg to 7 $? Why should that work for vectors? Or are you talking about polar and axial vectors? $\endgroup$ – pfnuesel Sep 14 '14 at 14:52
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    $\begingroup$ What would it mean to have added acceleartion and velocity. What physical quantity could that possible correspond to. Note, however, theat these consideration have nothing to do with the scalar of vector nature of the quantities and everything to do with what the quantities are (i.e. units). $\endgroup$ – dmckee Sep 14 '14 at 14:53
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    $\begingroup$ A direct quote from your book would be nice. I suspect you are misreading. $\endgroup$ – David Hammen Sep 14 '14 at 15:06
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    $\begingroup$ It would be very good if you clarified the question. I still suspect you are misreading. What is the name of the book? The name of the class? I suspect the answer to both questions is "Linear Algebra." $\endgroup$ – David Hammen Sep 14 '14 at 15:55
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    $\begingroup$ To put it simply, you can only add quantities with the same units, example, weight to weight (or mass to mass), time to time, money to money, etc. It is the same thing with vectors. Acceleration and velocity are both vectors, but with different units, so should not be added. $\endgroup$ – user13267 Sep 15 '14 at 2:23
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I doubt if your textbook makes it explicit, but the only sacred tenet in here is to respect dimensional homogeneity. One can make no sense of the sum of quantities with different dimensions.

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  • $\begingroup$ Check carefully, maybe it is implied somewhere that they refer to vectors having the same dimensions. This isn't the kind of a mistake that a textbook is expected to make. $\endgroup$ – 299792458 Sep 14 '14 at 15:01
  • $\begingroup$ ok i shall check $\endgroup$ – Jai Mahajan Sep 14 '14 at 15:04
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    $\begingroup$ I wouldn't say it's the only sacred tenet. A somewhat subtle lesson to take from GR is that tangent vectors at different points on a manifold can't be directly added (or subtracted). That's an instance of the more general point that you can only add vectors which belong to the same vector space - although one could argue that's circular reasoning since being addable is part of the definition of a vector space; still, the point is there are reasons for it not to work other than a unit mismatch. $\endgroup$ – David Z Sep 15 '14 at 2:46
  • $\begingroup$ @DavidZ - I know your comment and (the other) David's answer are more general than mine, but the OP is apparently a schoolkid. Let's have mercy on him :P No, but your point is valid. Thanks :) $\endgroup$ – 299792458 Sep 15 '14 at 4:21
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    $\begingroup$ @New_new_newbie true, but at the same time part of what we mean when we say this is an "expert-level" site is that we shouldn't feel obliged to refrain from giving advanced information when it's relevant. And in this case I think your answer oversimplifies the situation a bit too much - though not enough to be downvote-worthy or anything like that. $\endgroup$ – David Z Sep 15 '14 at 4:32
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From the commentary to the question, the textbook in question appears to be a mathematics textbook rather than a physics textbook. In mathematics, any two elements of a vector space can be added to one another to yield another member of that space. This is one of the requisites of what it means for something to be a "vector" in mathematics. Specifically, a set of objects forms a vector space over a field if

  • Addition amongst elements of the set is well defined (i.e., any two elements of the vector space can be added to one another, with the result being a member of the vector space),
  • Vector addition is commutative and associative,
  • The set contains an additive element, the zero vector, which when added to any other element yields that other element,
  • Every element of the set has an additive inverse,
  • Scalar multiplication, multiplication of elements of the vector space by members of the field is well defined (i.e., any element of the vector space be multiplied by any member of the field, with the result being a member of the vector space),
  • Scalar multiplication is associative and distributive, and
  • Multiplying any vector by the multiplicative identity of the field yields the original vector.

Mathematicians typically don't worry about units. When they do, they would deem the space of velocity vectors and the space of acceleration vectors to be two very distinct spaces.

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  • $\begingroup$ ok thanks for the info but this is a physics book of class 11. i dont think the question makers would have gone into this much depth $\endgroup$ – Jai Mahajan Sep 14 '14 at 16:37
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    $\begingroup$ I've asked you many times to clarify the question. Your comments to the question make it appear that this is exactly the context in which vectors are described in that text. When you ask if something is a misprint, it is best to give a full and exact quote, and to give some context. Do not use your interpretation of what the author wrote. Use the author's own words. $\endgroup$ – David Hammen Sep 14 '14 at 16:42
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    $\begingroup$ Good answer. Another way of putting it (with almost your words): "You can add two vectors if they belong to the same space. If they have different units, they do not belong to the same space. Ergo..." $\endgroup$ – Floris Sep 14 '14 at 23:21
  • $\begingroup$ @Floris i need your help....pls can i ask you a question regarding an answer you posted on my previous question? $\endgroup$ – Jai Mahajan Sep 15 '14 at 4:12
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    $\begingroup$ @user166748 for anything that concerns an answer to your earlier question, you should comment on that answer, not this one. $\endgroup$ – David Z Sep 15 '14 at 4:33
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Can we add any two vectors? My book says it is true

For example, adding acceleration to velocity (seems impossible).

Quite simply, your book meant two vectors of the same type.

It's just that simple. As you thought, you can not add two vectors that are "different things"!

(If you're just getting started with vectors. Note that indeed vectors have a certain number of dimensions. Of course, you can't even add them at all if they have different dimensions!)

It's really just that simple. Much as you can't add "three apples and four oranges".

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  • $\begingroup$ +1. To be picky though, you can add three apples and four oranges: the answer is "seven pieces of fruit". $\endgroup$ – Harry Johnston Sep 15 '14 at 3:31
  • $\begingroup$ Hi Harry - that's a funny point, but the critical issue at hand is questions should be answered at the relevant level. For physicists in particular, this concept strikes home: if one is asking a question in the newtonian realm, say, it's incorrect to point out related concepts in the relativistic realm, or quantum realm, or, vice versa. $\endgroup$ – Fattie Sep 15 '14 at 5:46
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    $\begingroup$ For example, note that below, fibo has simply confused the OP, by referring to direct sum concepts. A shame! $\endgroup$ – Fattie Sep 15 '14 at 5:47
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You can ague that you can add any vector, since you can look at a adding vectors with different units as other dimensions. So you example of adding velocity and acceleration, both in three spacial dimensions, will give you a six dimensional vector. An example of this would be phase space.

However usually those vectors with higher dimensions do not have any physical meaning, so in most formula the units of scalars and vectors you would add together will be the same.

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  • $\begingroup$ so we can add a velocity vector to a angular frequency vector? $\endgroup$ – Jai Mahajan Sep 14 '14 at 15:33
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    $\begingroup$ @user166748 yes, but it probably will not have any physical meaning, it just that mathematicaly it is possible by using more dimensions. $\endgroup$ – fibonatic Sep 14 '14 at 15:36
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    $\begingroup$ Right, fibonatic's answer is essentially referring to the direct sum construction. $\endgroup$ – Qmechanic Sep 14 '14 at 15:42
  • $\begingroup$ While technically correct, I don't know that this is what anyone means when they talk about adding vectors (at least, 99% of the time, and almost surely not in this question). $\endgroup$ – David Z Sep 15 '14 at 2:48
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    $\begingroup$ @user166748 , you can not add velocity and angular frequency. It's Just That Simple. Fibonatic is talking about a competely unrelated mathematical issue. Note, user166748, you don't have to worry about vectors AT ALL in this question: think of your car: consider the "speed" (180 km/h) and the "mass" (1800 kg). Can you add those things? Answer -- NO. Honestly it's just that simple. Don't forget that "one dimensional" things (eg "speed", "mass") are indeed also vectors! ("1 dimensional vectors!") Your question is honestly that simple. The textbook meant the same type of things. $\endgroup$ – Fattie Sep 15 '14 at 5:49

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