I'm wondering that why people doesn't care about vector's location. but, when we add the vectors, we move two or one vector to unite the vector's start point. and this mean location is important. but, when we learn very first of vector, we learn the vector doesn't matter where it is. in this point, I'm so confused because I can't find the reason that why vector's location doesn't matter. so i want to ask you - why the vector's location is not important?
thank you
On
In geometry, a vector represents a translation. In the length, direction and orientation of the vector is the complete information which determines the translation. In this approach, the position of the vector really doesn't matter.
It is not so in mechanics, where the position is equally important.
Thus when dealing with vectors you need to know which "field" they concern.
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The confusion is maybe due to some possible double meaning in the notation. Consider for instance:
$V=\begin{pmatrix} 1 \\ 0 \\ 1 \end{pmatrix} $
In essence this double meaning only occurs when picking coordinates to represent a vector. Clearly a 3-tuple of numbers can be considered as a point in 3 dimensional space. However, it can also be thought of as a direction, i.e. line of length $\sqrt{2} $ starting at the origin. Generally speaking if we only consider(or care about) the data of length and direction, the point from which the direction comes from is of little importance, hence there should be an equivalence of vectors by translation. In fact, where i am from, it was back in the day not so uncommon to introduce vectors as just this data. I.e. a length and a direction with equivalence relation given by translation, and then afterwards introduce coordinate systems to describe the operations precisely. Furthermore, translation invariance is great for physics and geometry since directions, e.g. tangent directions and forces acting on objects in space have meaning not as a direction from the origin, but really at certain points.
I guess one could be more specific when one wants a concrete base point of the vector by including it in the notation (one does so for instance when one considers “vector fields“ in physics, describing how forces act and varies at points, or for instance in differential geometry, when one considers vector spaces attached to certain points of a space, giving a model for the directions and “speed” of curves passing through that point). However, in a standard vector space, the connection between the nice algebraic operations in a vector space, and the arguably even nicer geometric interpretation of vectors would be lost if we do not require translation invariance.
For instance, in the usual interpretation of adding the vectors $v_{x}+w_{x+v}$(where $_{x} $ means the starting point of the vector ) could be defined algebraically and geometrically. However, we do NOT have commutativitity, and we cannot even make sense of$w_{x+v}+v_{x}$. Even worse, we cannot even rescale the $v_{x}$ vector by a number b and adding the other vector to it, since the endpoint of $bv_{x} $ would not match up with w_{x+v}.
I hope this was of some help
What you are probably getting at are 'free vectors'. Their description in school math is usually something like $\vec{x} = (a,b)$ - that means, go right $a$ units and up $b$ units. Ok, but starting from where?? It doesn't matter. That is the meaning of 'freedom' in this sense. You can 'freely' pick the point on the plane from which you apply the vector $\vec{x}$, for instance the origin.
We also have vectors called 'bound vectors' for which location does matter.