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In the product $\vec{F}=q(\vec{v}\times\vec{B})=q\vec{v}\times(B\hat{i}+B\hat{j}+B_0\hat{k})$. For $q=1$ and $\vec{v}=2\hat{i}+4\hat{j}+6\hat{k}$ and $\vec{F}=4\hat{i}-20\hat{j}+12\hat{k}$. What will be the complete expression for $\vec{B}$?
A
$-6\hat{i}-6\hat{j}-8\hat{k}$
B
$8\hat{i}+8\hat{j}-6\hat{k}$
C
$6\hat{i}+6\hat{j}-8\hat{k}$
D
$-8\hat{i}-8\hat{j}-6\hat{k}$
Detailed Solution
$\vec{F} = q(\vec{v} \times \vec{B})$ with q = 1:
$4\hat{i} - 20\hat{j} + 12\hat{k} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 2 & 4 & 6 \\ B & B & B_0 \end{vmatrix}$
Comparing components: $4 = 4B_0 - 6B$, $-20 = -2B_0 + 6B$, $12 = 2B - 4B$
Solving: B = –6, $B_0 = -8$
$\vec{B} = -6\hat{i} - 6\hat{j} - 8\hat{k}$