The vector $\overline{\mathrm{i}}+\mathrm{x}\overline{\mathrm{j}}+3\overline{\mathrm{k}}$ is rotated through an angle $\mathrm{\theta }$ and doubled in magnitude. It now becomes $4\overline{\mathrm{i}}+(4\mathrm{x}-2)\overline{\mathrm{j}}+2\overline{\mathrm{k}}$. The values of x are

We have denote $\overline{{\mathrm{v}}_1}=\overline{\mathrm{i}}+\mathrm{x}\overline{\mathrm{j}}+3\overline{\mathrm{k}},{\overline{\mathrm{v}}}_2=4\overline{\mathrm{i}}+(4\mathrm{x}-2)\overline{\mathrm{j}}+2\overline{\mathrm{k}}$.  then according to the given condition, we write $\left|{\overrightarrow{\mathrm{v}}}_2\right|=2\left|{\overline{\mathrm{v}}}_1\right|$then we have to substitute the magnitudes of ${\overline{\mathrm{v}}}_1 \mathrm{and} \overline{{\mathrm{v}}_2}$. in this equation and solve for $\mathrm{x},\mathrm{x}=-\frac{2}{3}$