Question

Two vectors $\overrightarrow{A}$ and $\overrightarrow{B}$ inclined at an angle $\theta$ have a resultant $\overrightarrow{R}$ which makes an angle $\alpha$ with $\overrightarrow{A}$ and angle $\beta$ with $\overrightarrow{B}$. Let the magnitudes of the vectors $\overrightarrow{A}$, $\overrightarrow{B}$ and $\overrightarrow{R}$ be represented by A, B and R respectively. Which of the following relations is not correct?

• A. $Asin\alpha =Bsin\beta$
• B. $Rsin\alpha =Bsin(\alpha +\beta )$
• C. $Rsin\beta =Asin(\alpha +\beta )$
• D. None of these

Answer 1

Answer: (C)

$Rsin\beta =Asin(\alpha +\beta )$

Let OP and OQ represent two vectors $\overrightarrow{A}$ and $\overrightarrow{B}$ making an angle $(\alpha +\beta )$. Using the parallelogram method ofvector addition,
Resultant vector, $\overrightarrow{R}=\overrightarrow{A}$ + $\overrightarrow{B}$
SN is normal to OP and PM is normal to OS.
From the geometry of the figure,
$O{S}^2=O{N}^2+S{N}^2=(OP+PN{)}^2+S{N}^2$= $(A+Bcos(\alpha +\beta ){)}^2+(Bsin(\alpha +\beta ){)}^2$

$R^2=A^2+B^2+2ABcos(\alpha +\beta )$

In $\Delta OSN,SN=OSsin\alpha =Rsin\alpha$ and in $\Delta PSN,SN=PSsin(\alpha +\beta )=Bsin(\alpha +\beta )$

$Rsin\alpha =Bsin(\alpha +\beta )$ or R/$sin(\alpha +\beta )$= B/$sin\alpha$
Similarly,

$PM=Asin\alpha =Bsin\beta$
$A/sin\beta =B/sin\alpha$
Combining (i) and (ii), we get

$R/sin(\alpha +\beta )=A/sin\beta =B/sin\alpha$
From eqn, (iii), $Rsin\beta =Asin(\alpha +\beta )$