Question

The resultant of $P$ and $Q$ is $R$. If $Q$ is doubled, $R$ is also doubled and if $Q$ is reversed, $R$ is again doubled. Then, $P^2:Q^2:R^2$ given by

• A. $2:2:3$
• B. $3:2:2$
• C. $2:3:2$
• D. $2:3:1$

Answer 1

The correct option is C $2:3:2$

Given, $R=P+Q$
$|R{|}^2=|P{|}^2+|Q{|}^2+2|P||Q|\cos \theta ...\left(1\right)$
Now $2R=P+2Q$
$4|R{|}^2=|P{|}^2+4|Q{|}^2+4|P||Q|\cos \theta ...\left(2\right)$
Now $2R=P-Q$
$4|R{|}^2=|P{|}^2+|Q{|}^2-2|P||Q|\cos \theta ...\left(3\right)$
Add $\left(1\right)$ and $\left(3\right)$
$5|R{|}^2=2|P{|}^2+2|Q{|}^2...\left(4\right)$
Now multiply equation $\left(3\right)$ with $2$ and then add it with $\left(2\right)$
$12|R{|}^2=3|P{|}^2+6|Q{|}^2$
$\Rightarrow 4|R{|}^2=|P{|}^2+2|Q{|}^2...\left(5\right)$
Now equation $\left(4\right)$-$\left(5\right)$
$|R{|}^2=|P{|}^2$
Now from equation $\left(4\right)$
$5|R{|}^2=2|R{|}^2+2|Q{|}^2$
$\Rightarrow |Q{|}^2=\frac{3}{2}|R{|}^2$
Therefore, $|P{|}^2:|Q{|}^2:|R{|}^2=|R{|}^2:\frac{3}{2}|R{|}^2:|R{|}^2$
$=2:3:2$