A vector perpendicular to the vector $\left(\hat{\mathrm{i}}+2\hat{\mathrm{j}}\right)$ and having magnitude $3\sqrt{5}$ units is

Let the vector be $\mathrm{x}\hat{\mathrm{i}}+\mathrm{y}\hat{\mathrm{j}}$.

as the vector is perpendicular to $\left(\hat{\mathrm{i}}+2\hat{\mathrm{j}}\right)$, the dot product is zero.

we get, $\mathrm{x}+2\mathrm{y}=0$.

As given the magnitude of vector is $3\sqrt{5}$.

${\mathrm{x}}^2+{\mathrm{y}}^2=45$.

on solving we get, the vector is $6\hat{\mathrm{i}}-3\hat{\mathrm{j}} \mathrm{or} -6\hat{\mathrm{i}}+3\hat{\mathrm{j}}$.