How & why does the law of vector addition work? Our teacher explained vector addition to us. He explained to us the triangle law of vector Addition.
I have two questions:


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*He said the vector $\vec{R}$ is the resultant vector, which means that instead of going through $\vec{A}$ and $\vec{B}$ we could have directly gone through $\vec{R}$. I don't really understand it and obviously it is easy to understand this intuitively when talking about displacement. But bringing forces and other vector quantities into this image is really hard to understand. So how does the triangle law work for adding forces?

*The head-tail combining rule is confusing and appears as a trick to enable memorization. So if there is a triangle law of vector addition, then why is there a need for the parallelogram law Of vector addition, when both are talking about adding entities with directions and magnitudes?
Also, how does one understand force vector addition using the parallelogram law (I believe this question will be answered when the first question is answered)?
Edit: I am new here, so I don't know how to add images to the question. Also please don't answer this question mathematically, please answer in a way that vector additions make sense intuitively and become easy to imagine.
 A: As you say, it is easy to understand when talking about displacement. 

Then you only need to recognise that a displacement vector is the same vector wherever it appears on the plane. 

(this also shows the parallelogram law. This is not different from the triangle law, just a different way of thinking of it). 
So, to apply this to forces, and other vector quantities, you only have to recognise that it does not matter where you put the vectors on the diagram. Of course, it only applies to questions where only the vector properties of the forces are relevant -- like when you are calculating the net force acting in a particular direction. It does not apply when you need to take moments, and have to describe the force acting at a point. 
A: The triangle law of vector addition is the coordinate-free definition of vector addition, so one cannot explain why it describes addition because it is addition.
Vectors are geometric objects, regardless of what you are describing. If force is a vector, then its coordinate free definition of addition is the same as displacement vectors, or electric field strengths, or any other vector.
The only thing the parallelogram law adds is that it explicitly shows that addition is commutative:
$$\vec A + \vec B = \vec B + \vec A$$
