A vector $\overrightarrow{\mathrm{a}}$ has the components $2p$ and $1$ w.r.to a rectangular Cartesian system This system is rotated through a certain angle about the origin in the counter clockwise sense If, with respect to a new system, $\overrightarrow{\mathrm{a}}$ has components$(p+1)$ and $1,$then $p=$

$$\begin{gathered} \text{ We have, }\overrightarrow{a}=2p\widehat{i}+\widehat{j}\text{. } \\ \text{ Let }\overrightarrow{b}\text{ be the vector obtained from }\overrightarrow{a}\text{ by rotating the axes. Then, } \end{gathered}$$

$\text{ the components of }\overrightarrow{b}\text{ are }p+1\text{ and }1\text{. Therefore, }$

$\overrightarrow{b}=(p+1)\widehat{\alpha }+\widehat{\beta }\text{ where }\widehat{\alpha }\text{ and }\widehat{\beta }\text{ are unit vector along the new axes. }$

$\begin{array}{l}\left|\overrightarrow{b}\right|=\left|\overrightarrow{a}\right|\\ 4p^2+1=(p+1{)}^2+1\Rightarrow 3p^2-2p-1=0\Rightarrow p=1,-1/3\end{array}$