Understanding Parallelogram Law of Vector Addition? Recently I've been adding vectors using the Parallelogram Law and the maths is trivial. However, I can't understand the underlying principals. What allows us to move a vector such that the tail meets the head of the other vector? Why can we move the vector to a new starting position like this. Furthermore why does the Parallelogram Law hold in general. Is there some intuition or proof behind this theorem?
 A: Strictly speaking, adding vectors using the Parallelogram Law doe not require any movement of a vector such that the tail meets the head of the other vector. Such a procedure involving movement should deserve a different name.
Parallelogram Law has such a name because representing two vectors as oriented segments, having their tails at the same point, defines a parallelogram whose diagonal passing through the common point can represent the sum of the two vectors. Of course, in a Euclidean space, the same resulting vector could be described as the end point of a path made by the first oriented segment, followed by another oriented segment whose tail starts at the head of the first one. The latter description does not explicitly introduce a parallelogram.
The two descriptions correspond to two related but different concepts of vector space  and affine spaces, at the origin of the concepts of bound and free vectors.
Regarding the explicit questions, both versions of the sum of two vectors, the bound (aka vector space, Parallelogram Law version) and the free (aka affine space, path-of-displacement version), cannot be proved because they are definitions. What can be proved, but it is a trivial exercise of Euclidean geometry, is that they are consistent.
The possibility of moving vectors exists only in affine spaces. However, since every vector space may be considered an affine space over itself, the distinction between the two concepts is often blurred. In Physics, it is quite usual to use both implicitly.
One of the best resources I know making clear the distinction between vectors in a vector space and Euclidean affine spaces is the compact but very clear booklet by A. Lichnerowicz Elements of Tensor Calculus. Some pdf file is also present on the Internet.
A: Try thinking about the one-dimensional case:  How do you add the numbers 3 and 4?  You move along the number line from 0 to 3, and then, starting at 3, you move another 4 units, landing at 7.
Likewise, to add two vectors of any dimension, you start by adding the first vector to zero, which brings you to that vector's "head", then starting from there travel along the second vector to get your total.
A: I am not sure if you refer to this. Excuse how unrealistic the examples are. :)
Three men have 1 floating boat and a tractor each.
The first one moves his boat with the tractor first and then decides he wants to pull the boat perpendicularly.

The second man chooses to pull by hand the boat first, and after he gets tired uses the tractor.

The third one hops into the tractor and pulls a rope at the same time the tractor moves.

The result is the same, regardless you put one force after the other, or both are applied at the same time.
It is an oversimplification, but that is the idea. Remember that formulas and schematics are to simplify our calculations.
A: We don't physically move the vectors while adding them. Moreover, the parallelogram law of vectors is more of a "mathematical construct" which is used to model nature (for eg: Forces and velocity).
To be more specific, there are actual proofs for the parallelogram law of vectors but those proofs are purely mathematical. However, those arguments can also be extended to physics also because they seem to be very efficient tools for modelling nature.
As I already mentioned in the comments, the exact reason why it works for modelling forces at play is empirical and cannot be reduced to Newton's first principles.
Source: https://en.m.wikipedia.org/wiki/Parallelogram_of_force
A: You can think of the 2 vectors being components of the resultant vector , I like to think of it in the reverse way , The resultant vector's direction is given by the angle and its magnitude is given by how much units it has acquired(from both the vectors).  hence when you draw the diagrams for the resultant vector being divided by these 2 vectors you will get a parallelogram shape ! and moving a vector to a new starting position wont change anything about the vector because its magnitude and direction has not changed just the position (like shifting of origin of parabola wont change the parabola's nature!)
A: There is an important mathematical notion that it is necessary to use: the congruence or isometry, in particular, translation.
The axioms of the congruence allow to define a movement in the Euclidean geometry but implicitly, on the other hand Hilbert introduces the congruence as a basic notion.
In Euclid's sense, motion is an obvious intuitive notion, that it is useless to explain by a special axiom.
the figures which coincide are considered as equal.
The motion of a figure, or of the whole space, takes the name of translation or rotation under certain mathematical conditions and it is shown that the composition of several motions is a motion and this theorem is called a group property (which led to the Erlangen program).
There is another implicit notion that the binary relation and in particular the binary relation of equivalence (reflexive+ symmetric+ transitive).
Theorem: Let $A$ be a set of oriented segments of a given plane. The equipolence relation of the oriented segments is an equivalence relation on $A$.
A: 
to add the components of two vectors you must chose coordinate system.

thus:
I)
\begin{align*}
 &\vec{V}_{14}=\vec{V}_{13}+\vec V_{12}= \begin{bmatrix}
            x_{13} \\
            0\\
          \end{bmatrix} +
           \begin{bmatrix}
            x_{12} \\
            y_{12} \\
          \end{bmatrix} =       
     \begin{bmatrix}
            x_{13}+x_{12} \\
            y_{12} \\
          \end{bmatrix}   \tag 1   
  \end{align*}
II)

we shift the vector $~\vec V_{12}~$ parallel to point 3 and obtain $~\vec V_{1'2'}~$. At point 3 we put additional coordinate system $~S'~$. thus
\begin{align*}
&\left(\vec{V}_{14}\right)_S=\left(\vec{V}_{11'}\right)_S+\left(\vec V_{1'2'}\right)_{S'}= \begin{bmatrix}
            x_{13} \\
            0\\
          \end{bmatrix} +
           \begin{bmatrix}
            x_{12} \\
            y_{12} \\
          \end{bmatrix} =       
     \begin{bmatrix}
            x_{13}+x_{12} \\
            y_{12} \\
          \end{bmatrix}     
  \end{align*}
so you  obtain also the result of equation (1)
A: It can be thought as the viewpoint of the differential geometry.
Think in curved space (manifold). At every point $p$ of that curved space we can construct a space where living all the tangent vectors of the manifold at the point $p$, this space is called the tangent space $T_p$. In the case of a sphere, for example, imagine that in the north pole we can construct an imaginary plane containing all tangent vectors of the sphere in that point $T_p$. But if we move to the equator, the tangent space of the sphere in any point of the equator $T_q$ will not be the same that the tangent space in the north pole $T_p$. Indeed, this spaces are orthogonal.
Now, imagine that we want to compare two vectors of these different tangent spaces (where by "comapare" we mean add, subtract, take the dot product, and so on). What we must do, is to take one vector, transport-parallel it to the the tangent space of the other. The thing is that in Euclidean flat space (the one of a lifetime), to move a vector from one point to another, while keeping it constant, because the tangent vector of a plane is the same plane, therefore, the vectors that we campare in the flat space don't need to be parallel trasnported because they already live in the same tangent space. This is the reason whiy it is natural in flat space, to move a vector from one point to another while keeping it constant. This concept  of moving a vector along a path, keeping it constant all the while, is known as parallel transport.
A: According to definition of a vector it is defined as a quantity which have magnitude as well as direction
The reason we can move a vector in this way is because we are not changing its magnitude or direction(essentially the vector remains same). In other words, the vector is simply being translated, or moved to a new position in space, without being rotated or scaled in any way.
