Question

Two vectors $\overrightarrow{A}and\overrightarrow{B}$ have equal magnitude. The magnitude of $(\overrightarrow{A}+\overrightarrow{B})$ is 'n' times the magnitude of $(\overrightarrow{A}+\overrightarrow{B})$. The angle between $\overrightarrow{A}+\overrightarrow{B}$ is '

• A. $si{n}^{-1}\left[\frac{n^2-1}{n^2+1}\right]$
• B. $co{s}^{-1}\left[\frac{n-1}{n+1}\right]$
• C. $co{s}^{-1}\left[\frac{n^2-1}{n^2+1}\right]$
• D. $si{n}^{-1}\left[\frac{n-1}{n+1}\right]$

Answer 1

Answer: (C)

$co{s}^{-1}\left[\frac{n^2-1}{n^2+1}\right]$

According to the questions:
$\Rightarrow A^2+A^2+2ABcos\theta =n^2(A^2+A^2-2ABcos\theta )$

$\Rightarrow A^2+A^2+2A^{2}cos\theta =n^2(A^2+A^2-2A^{2}cos\theta )$

$\Rightarrow 2A^2(1+cos\theta )=2A^2{n}^2(1-cos\theta )$

$\Rightarrow 2+cos\theta =n^2-n^{2}cos\theta .$

$\Rightarrow cos\theta (1+n^2)=n^2-1$

$\Rightarrow cos\theta =\frac{n^2-1}{n^2+1}$

$\Rightarrow \theta =co{s}^{-1}\left(\frac{n^2-1}{n^2+1}\right)$
Hence, the correct option is $\left(C\right)$