A charged particle carrying charge $1 \mu \mathrm{C}$ is moving with velocity $(2 \hat{\mathrm{i}}+3 \hat{\mathrm{j}}+4 \hat{\mathrm{k}}) \mathrm{ms}^{-1}$. If an external magnetic field of $(5 \hat{\mathrm{i}}+3 \hat{\mathrm{j}}-6 \hat{\mathrm{k}}) \times 10^{-3} \mathrm{~T}$ exists in the region where the particle is moving then the force on the particle is $\overrightarrow{\mathrm{F}} \times 10^{-9} \mathrm{~N}$. The vector $\vec{F}$ is :
Correct Option: , 3
$\overrightarrow{\mathrm{F}}=9(\overrightarrow{\mathrm{V}} \times \overrightarrow{\mathrm{B}})$ (Force on charge particle moving
in magnetic field)
$\overrightarrow{\mathrm{V}} \times \overrightarrow{\mathrm{B}}=(2 \hat{\mathrm{i}}+3 \hat{\mathrm{j}}+4 \hat{\mathrm{k}}) \times(5 \hat{\mathrm{i}}+3 \hat{\mathrm{j}}-6 \hat{\mathrm{k}}) \times 10^{-3}$
$=\left(\begin{array}{ccc}\hat{i} & \hat{j} & \hat{k} \\ 2 & 3 & 4 \\ 5 & 3 & -6\end{array}\right) \times 10^{-3}$
$=[\hat{\mathrm{i}}[-18-12]-\hat{\mathrm{j}}[-12-20]+\hat{\mathrm{k}}[6-15]] \times 10^{-3}$
$=[\hat{\mathrm{i}}[-30]+\hat{\mathrm{j}}[32]+\hat{\mathrm{k}}[-9]] \times 10^{-3}$
Force $=10^{-6}[-30 \hat{\mathrm{i}}+32 \hat{\mathrm{j}}-9 \hat{\mathrm{k}}] \times 10^{-3}$
$=10^{-9}[-30 \hat{\mathrm{i}}+32 \hat{\mathrm{j}}-9 \hat{\mathrm{k}}]$
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