To solve this problem, we need to find the magnetic field vector $\vec{B}$ such that the force $\vec{F}$ is given by the equation:$\\ \vec{F} = q (\vec{v} \times \vec{B}) \\$Given:$\\ q = 1, \\ \vec{v} = 2\hat{i} + 4\hat{j} + 6\hat{k}, \\ \vec{F} = 4\hat{i} - 20\hat{j} + 12\hat{k}, \\ \vec{B} = B_x \hat{i} + B_y \hat{j} + B_0 \hat{k} \\$We need to compute the cross product $\vec{v} \times \vec{B}: \\ \vec{v} \times \vec{B} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 2 & 4 & 6 \\ B_x & B_y & B_0 \\ \end{vmatrix} \\$Expanding the determinant, we get:$\\ \vec{v} \times \vec{B} = \hat{i}(4B_0 - 6B_y) - \hat{j}(2B_0 - 6B_x) + \hat{k}(2B_y - 4B_x) \\$Equating this to the given force $\vec{F} = 4\hat{i} - 20\hat{j} + 12\hat{k},$ we have:$\\ 4B_0 - 6B_y = 4 \quad$(1)$\\ 2B_0 - 6B_x = 20 \quad$(2)$\\ 2B_y - 4B_x = 12 \quad$(3)$\\$Solving these equations:$\\$From equation (1):$\\ 4B_0 - 6B_y = 4 \\ B_0 = \frac{4 + 6B_y}{4} \\ B_0 = 1.5 + 1.5B_y \quad$(4)$\\$From equation (2):$\\ 2B_0 - 6B_x = 20 \\ B_0 = 10 + 3B_x \quad$(5)$\\$Equating equations (4) and (5):$\\ 1.5 + 1.5B_y = 10 + 3B_x \\ 1.5B_y = 8.5 + 3B_x \\ B_y = \frac{8.5 + 3B_x}{1.5} \\ B_y = \frac{17 + 6B_x}{3} \quad$(6)$\\$Substitute $B_y$ from equation (6) into equation (3):$\\ 2\left(\frac{17 + 6B_x}{3}\right) - 4B_x = 12 \\ \frac{34 + 12B_x}{3} - 4B_x = 12 \\ 34 + 12B_x - 12B_x = 36 \\ 34 = 36 \quad$(This is incorrect, indicating a calculation error)$\\$Re-evaluate the equations:$\\$From equation (1):$\\ 4B_0 - 6B_y = 4 \\ B_0 = 1 + 1.5B_y \\$From equation (2):$\\ 2B_0 - 6B_x = 20 \\ B_0 = 10 + 3B_x \\$Equate the two expressions for $B_0: \\ 1 + 1.5B_y = 10 + 3B_x \\ 1.5B_y = 9 + 3B_x \\ B_y = 6 + 2B_x \\$Substitute $B_y = 6 + 2B_x$ into equation (3):$\\ 2(6 + 2B_x) - 4B_x = 12 \\ 12 + 4B_x - 4B_x = 12 \\ 12 = 12 \quad$(This is consistent)$\\$Now, solve for $B_x: \\ B_y = 6 + 2B_x \\ B_0 = 10 + 3B_x \\$Using the options, substitute and verify:$\\$Option 2: $-6\hat{i} - 6\hat{j} - 8\hat{k} \\ B_x = -6, \\ B_y = -6, \\ B_0 = -8 \\$Check:$\\ B_y = 6 + 2(-6) = 6 - 12 = -6 \\ B_0 = 10 + 3(-6) = 10 - 18 = -8 \\$This satisfies all equations, so the correct option is Option 2.