What is the analytical treatment of triangle law of vector addition?

Polygon law is applicable in both the situation given but the equation given in the reason is not useful in explaining the assertion.

From triangle law of addition, we can clearly write answer.

$\left|\overrightarrow{P}\right|=\left|\overrightarrow{Q}\right|=x$

let the angle between $\overrightarrow{P}and\overrightarrow{Q}$ $\stackrel{}{}\stackrel{}{}$ is , so according to question, we can write $\stackrel{}{}\stackrel{}{}\stackrel{}{}\stackrel{}{}\text{exception 25:}\text{exception 25:}$

^{$\left|\overrightarrow{P}+\overrightarrow{Q}\right|=n\times \left|\overrightarrow{P}-\overrightarrow{Q}\right|\Rightarrow \sqrt[]{P^2+Q^2+2PQcos\theta }=n\times \sqrt[]{P^2+Q^2-2PQcos\theta }$}

$\Rightarrow \sqrt[]{x^2+x^2+2x^{2}cos\theta }=n\times \sqrt[]{x^2+x^2-2x^{2}cos\theta }$

$\Rightarrow \theta =co{s}^{-1}\left(\frac{n^2-1}{n^2+1}\right)$

Graphically, we place vectors head-to-tail. Analytically, we add components along axes using coordinate geometry, trigonometric functions, and unit vectors for precise vector resultant calculation.