A vector a has components 2p and 1 w.r.t a rectangular cartesian system. This system is rotated through a certain angle about the origin in the counter-clockwise sense. If w.r.t the new system, a has components p+1 and 1, then

Let i, j be unit vectors along the co-ordinate axes
On rotation, let b be the vector having components p+1 and 1
where i,j are units vectors along the new co-ordinate axes.
As we know on rotation magnitude of the initial and final vector will remain same.
\(\left |\vec{b} \right |=\left |\vec{a} \right ||=\left |b^{2} \right |=\left |a^{2} \right |\)\((p+1)^{2}+1=(2p^{2})+1\)\((3p)^{2}-2p-1=0\)\((3p+1)(p-1)=0\)Hence p=1 or p=-1/3

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