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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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.
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
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.
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".
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.