A vector has components $$2p$$  and $$1$$ with respect to a rectangular cartesian system. The axes are rotated through an angle $$\alpha $$ about the origin in the anticlockwise sense. If the vector has components $$p + 1$$  and $$1$$ with respect to the new system then :

Solution :
Here, $$\overrightarrow a = 2p\overrightarrow i + \overrightarrow j .$$   After rotation, let the vector be $$\overrightarrow b $$ and let the unit vectors along the new axes be $$\overrightarrow {i'} ,\,\overrightarrow {j'} .$$
Then $$\overrightarrow b = \left( {p + 1} \right)\overrightarrow {i'} + \overrightarrow {j'} .$$     But the magnitude of a vector does not change with rotation of axes.
$$\eqalign{ & \therefore \,\,\,\overrightarrow {\left| a \right|} = \left| {\overrightarrow b } \right| \cr & \Rightarrow \sqrt {{{\left( {2p} \right)}^2} + {1^2}} = \sqrt {{{\left( {p + 1} \right)}^2} + {1^2}} \cr & \Rightarrow 4{p^2} + 1 = {p^2} + 2p + 2{\text{ or }}3{p^2} - 2p - 1 = 0 \cr & \therefore \,\,p = 1,\, - \frac{1}{3} \cr} $$