What is the reason behind triangle law of vector addition, in other words, how is this really justified?
A naïve answer...
If you go from A to B then from B to C, you can represent your displacement from A to B as an arrow and from B to C to another arrow. Clearly your displacement from A to C can be represented by an arrow going from A to C or by the two arrows already mentioned, placed with the tail of the second touching the head of the first. This is the rule for adding displacements and arguably it is self-evident. The rule can be extended to any number of displacements.
Velocity is displacement per unit time and so velocities must add as displacements.
Momentum and acceleration are defined in terms of velocity, so momenta and accelerations must add as displacements.
The argument can be extended, via Newton's second law, to forces and field strengths.
Well I finally realized that we are actually not adding two quantities but trying to find the resultant. Therefore the resultant must indeed be the third side of the triangle as it will give us the shortest distance between the two points, i.e the tail of one and the head of another. By definition the resultant is that vector which can have the very same effect as the other two vectors would provide together.