The position vector of A is pi¯+j¯. If A is rotated about O through an angle π6 in anti clock wisedirection. It coincides with B whose position vector i¯+qj¯.The value of p, q are

|OA¯|=|OB¯|Cos⁡π6=OA¯⋅OB¯|OA¯|⋅|OB¯|=(pi¯+j¯)⋅(i¯+qj¯)p2+11+q2⇒(p+q)2=p2+11+q234 ......(1)⇒p2+1=q2+1⇒p=q⇒(2) Solving (1)and (2) ⇒3p4−10p2+3=0⇒3p2−1p2−3=0

The position vector of A is pi¯+j¯. If A is rotated about O through an angle π6 in anti clock wisedirection. It coincides with B whose position vector i¯+qj¯.The value of p, q are

|OA¯|=|OB¯|Cos⁡π6=OA¯⋅OB¯|OA¯|⋅|OB¯|=(pi¯+j¯)⋅(i¯+qj¯)p2+11+q2⇒(p+q)2=p2+11+q234 ......(1)⇒p2+1=q2+1⇒p=q⇒(2) Solving (1)and (2) ⇒3p4−10p2+3=0⇒3p2−1p2−3=0

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The position vector of A is $p \bar{i} + \bar{j}$ . If A is rotated about O through an angle $\frac{π}{6}$ in anti clock wise
direction. It coincides with B whose position vector $\bar{i} + q \bar{j}$ .The value of p, q are

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$- \sqrt{3}, \sqrt{3} or \frac{- 1}{\sqrt{3}}, \frac{1}{\sqrt{3}}$

$\frac{1}{3}, \frac{1}{3}$

$\sqrt{3}, 3$

$\sqrt{3}, \frac{1}{3}$

answer is C.

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Detailed Solution

$| \bar{O A} | = | \bar{O B} |$

$Cos \frac{π}{6} = \frac{\bar{O A} \cdot \bar{O B}}{| \bar{O A} | \cdot | \bar{O B} |} = \frac{(p \bar{i} + \bar{j}) \cdot (\bar{i} + q \bar{j})}{\sqrt{p^{2} + 1} \sqrt{1 + q^{2}}}$