In the product \(\vec{F} = q(\vec{v} \times \vec{B}) = q \vec{v} \times (B_x \hat{i} + B_y \hat{j} + B_0 \hat{k})\), 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 \(vec{B}\)?
\(6\hat{i} + 6\hat{j} - 8\hat{k}\)
\(-8\hat{i} - 8\hat{j} - 6\hat{k}\)
\(-6\hat{i} - 6\hat{j} - 8\hat{k}\)
\(8\hat{i} + 8\hat{j} - 6\hat{k}\)
Solution:
Using the relation \(\vec{F} = \vec{v} \times \vec{B}\), we compare vector components: \(4\hat{i} - 20\hat{j} + 12\hat{k} = (4 B_0 - 6 B_y)\hat{i} - (2 B_0 - 6 B_x)\hat{j} + (2 B_y - 4 B_x)\hat{k}\). Testing the values in Option C gives \(B_x = -6\, B_y = -6\, B_0 = -8\), which completely satisfies all equations.
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