Question

The position vectors of four points $P,Q,R,S$ are $2\overline{a}+4\overline{c},5\overline{a}+3\sqrt{3}\overline{b}+4\overline{c}),-2\sqrt{3}\overline{b}+\overline{c}$ and $2\overline{a}+\overline{c}$ respectively, then

• A. $\overline{PQ}$ is parallel to $\overline{RS}$
• B. $\overline{PQ}$ is not parallel to $\overline{RS}$
• C. $\overline{PQ}$ is equal to $\overline{RS}$
• D. $\overline{PQ}$ is parallel and equal to $\overline{RS}$

Answer 1

The correct option is A $\overline{PQ}$ is parallel to $\overline{RS}$

$P(2\overrightarrow{a}+4\overrightarrow{c})$ $Q(5\overrightarrow{a}+3\sqrt{3}\overrightarrow{b}+4\overrightarrow{c})$

$R(-2\sqrt{3}\overrightarrow{b}+\overrightarrow{c})$ $S(2\overrightarrow{a}+\overrightarrow{c})$

$\overrightarrow{PQ}=\overrightarrow{OQ}-\overrightarrow{OP}=(5\overrightarrow{a}+3\sqrt{3}\overrightarrow{b}+4\overline{c})-(2\overrightarrow{a}+4\overrightarrow{c})$

$\overrightarrow{PQ}=(3\overline{a}+3\sqrt{3}\overline{b})=3(\overrightarrow{a}+\sqrt{3}\overrightarrow{b})$

$\overrightarrow{RS}=\overrightarrow{OS}-\overrightarrow{OR}=(2\overline{a}+\overline{c})-(-2\sqrt{3}\overrightarrow{b}+c)$

$\overrightarrow{RS}=(2\overrightarrow{a}+2\sqrt{3}\overline{b})=2(\overrightarrow{a}+\sqrt{3}\overrightarrow{b})$

$\therefore \overrightarrow{PQ}=3(\overrightarrow{a}+\sqrt{3}\overrightarrow{b})\overrightarrow{RS}=(\overline{a}+\sqrt{3}b)$

Since directions of $PQ$ and $RS$ are same

$\overrightarrow{PQ}$ is parallel to $\overrightarrow{RS}$