$A$ vector $\vec{a}$ has components $2p$ and $1$ with respect to a two-dimensional rectangular Cartesian system. This system is rotated through a certain angle $\theta$ about the origin in the counter-clockwise direction. If $\vec{a}$ has components $p+1$ and $1$ with respect to the new system,then:

• A
  $p=1$ or $p=\frac{-1}{3}$
• B
  $p=-1$ or $p=\frac{1}{3}$
• C
  $p=1$ or $p=-1$
• D
  $p=0$ or $p=\frac{1}{2}$