A short magnetic needle is placed in a magnetic field $\vec{B}_i$ in the direction $(\sqrt{3}\hat{i}+\hat{j})$. The needle experiences a torque of 0.06 Nm. If the same magnetic needle is placed in a magnetic field $2\vec{B}_j$ in the direction $(\hat{i}+\sqrt{3}\hat{j})$, the torque experienced by it is
(Here $\vec{B}_i$ and $\vec{B}_j$ are unit vectors along x and y axes respectively. The text means magnitude $B$ in direction $(\sqrt{3}\hat{i}+\hat{j})$ and magnitude $2B$ in direction $(\hat{i}+\sqrt{3}\hat{j})$ assuming $B_i=B_j=B$ as magnitude factors. Or $B_i$ and $B_j$ are vector components, which is confusing with unit vectors $\hat{i}, \hat{j}$. The question most likely means initial field is $\vec{B}_1$ with magnitude $B_1$, and final field is $\vec{B}_2$ with magnitude $B_2$. Given the option format, let's assume the intended meaning is: Field 1 has magnitude $B$ and direction $\vec{u}_1 = \frac{\sqrt{3}\hat{i}+\hat{j}}{2}$. Field 2 has magnitude $2B$ and direction $\vec{u}_2 = \frac{\hat{i}+\sqrt{3}\hat{j}}{2}$. The phrasing "$\vec{B}_i$ in the direction..." and "$2\vec{B}_j$ in the direction..." is very confusing. Let's assume $B_i$ refers to some magnitude $B_0$ for the first field, and $B_j$ refers to the same magnitude $B_0$ for the second field, so the second field's magnitude is $2B_0$.
More standard interpretation: Initial field $\vec{B}_1$, magnitude $B_1$. Direction vector $\vec{d}_1 = \sqrt{3}\hat{i}+\hat{j}$. So $\vec{B}_1 = C_1 (\sqrt{3}\hat{i}+\hat{j})$.
Second field $\vec{B}_2$, magnitude $B_2$. Direction vector $\vec{d}_2 = \hat{i}+\sqrt{3}\hat{j}$. So $\vec{B}_2 = C_2 (\hat{i}+\sqrt{3}\hat{j})$.
The phrasing "magnetic field $\vec{B}_i$ in the direction $(\sqrt{3}\hat{i}+\hat{j})$" suggests the magnitude of this field is $B_i$. And "magnetic field $2\vec{B}_j$ in the direction $(\hat{i}+\sqrt{3}\hat{j})$" suggests the magnitude of this second field is $2B_j$. If $B_i$ and $B_j$ are just labels, not unit vectors, and if $B_i = B_j = B_{magnitude}$, then magnitudes are $B_{magnitude}$ and $2B_{magnitude}$. This seems most plausible.)
