Question

If $P , Q , R$ and $S$ are the points with position vectors $\hat{ i }+\hat{ j }-\hat{ k }, 2 \hat{ i }-\hat{ j }+3 \hat{ k }, 2 \hat{ i }-3 \hat{ k }$ and $3 \hat{ i }-2 \hat{ j }+\hat{ k }$ respectively, then the angle between $P Q$ and $R S$ is

• A. 0
• B. $\frac{\pi}{2}$
• C. $\pi$
• D. $\frac{\pi}{3}$

Answer 1

Answer: (A)

Given position vectors
$O P =\hat{ i }+\hat{ j }-\hat{ k }$
$O Q =2 \hat{ i }-\hat{ j }+3 \hat{ k }$
$O R =2 \hat{ i }-3 \hat{ k }$ and $O S =3 \hat{ i }-2 \hat{ j }+\hat{ k }$
So $P Q = O Q - O P =\hat{ i }-2 \hat{ j }+4 \hat{ k }$
and $R S = O S - O R =\hat{ i }-2 \hat{ j }+4 \hat{ k }$
As $\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}=\frac{c_{1}}{c_{2}}$
$\therefore $ Angle between $P Q$ and $RS$ is $0.$