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A vector $a$ has components 2 p and 1 with respect to a rectangular Cartesian coordinate system. This system is rotated through a certain angle about the origin in the counter clock –wise sense. If, with respect to new system, a has components p+1, and 1 then p=

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

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

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

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

answer is A.

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Detailed Solution

$A = (2 P, 1) B = (P + 1, 1) f r o m g i v e n d a t a | \vec{OA} | = | \vec{OB} | \Rightarrow 4 p^{2} + 1 = {(P + 1)}^{2} + 1 \Rightarrow 4 p^{2} = {(P + 1)}^{2} \Rightarrow p + 1 = 2 p o r p + 1 = - 2 p \Rightarrow p = 1, \frac{- 1}{3}$

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