Question

How did triangle law of vector addition originated? I mean how did scientist came up with the conclusion that if two vectors represent two regular sides of a triangle in same order, the third side of triangle in opposite order represents the resultant vector of these vectors? their is no history for development of this law.
Thanks in advance..

Answer 1

OK let's take it easy.
First you know that
let us take 2 vectors:
\[A =(A _{x}, A _{y}, A _{z})\]
\[B =(B _{x}, B _{y}, B _{z})\]
and rotate them to get \[A ^{/} =(A ^{/} _{x}, A ^{/} _{y}, A ^{/} _{z})\]
\[B ^{/} =(B ^{/} _{x}, B ^{/} _{y}, B ^{/} _{z})\]
If we let C be their sum
\[C = A+B =\left( A _{x}+B _{x},A _{y}+B _{y}, A _{z}+B _{z} \right)\]
\[C ^{/} = A^{/} +B^{/} =( A^{/} _{x}+B^{/} _{x},A^{/} _{y}+B ^{/} _{y}, A^{/} _{z}+B^{/} _{z})\]
thus they are the same according to the definition of the vector(symmetry).
Now we have just defined the sum operation in vector analysis.
The geometrical interpretation of this is the parallelogram rule.

Answer 2

Think of billiard balls. If you want to hit a ball, but another is in the way, you'll need to bank the shot. If another ball were not in the way, you would go straight at it. However, you can effectively accomplish the same result by banking the shot. The resulting parallelogram drawn by @Catch.me shows you why: