An electron moves through a uniform magnetic field $$\vec{B} = B_0\hat{i} + 2B_0\hat{j}$$ T. At a particular instant of time, the velocity of electron is $$\vec{u} = 3\hat{i} + 5\hat{j} \text{ m s}^{-1}$$. If the magnetic force acting on electron is $$\vec{F} = 5e\hat{k}$$ N, where $$e$$ is the charge of electron, then the value of $$B_0$$ is ____ T.

$$\vec{F} = q(\vec{v} \times \vec{B})$$

$$\vec{v} \times \vec{B} = (3\hat{i} + 5\hat{j}) \times (B_0\hat{i} + 2B_0\hat{j})$$

$$\implies \vec{v} \times \vec{B} = 3(2B_0)\hat{k} + 5B_0(-\hat{k}) = 6B_0\hat{k} - 5B_0\hat{k} = B_0\hat{k}$$

$$\vec{F} = 5e\hat{k}$$

$$5e\hat{k} = -e(B_0\hat{k})$$

$$\implies 5 = -B_0 \implies B_0 = -5\text{ T}$$ $$\implies |B_0| = 5$$