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(C) The plane passes through the point $(x_1, y_1, z_1) = (3, -2, -1)$ and is parallel to vectors $\vec{b} = (1, -2, 4)$ and $\vec{c} = (3, 2, -5)$.Th

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$2x + 17y + 8z + 36 = 0$

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(C) The plane passes through the point $(x_1, y_1, z_1) = (3, -2, -1)$ and is parallel to vectors $\vec{b} = (1, -2, 4)$ and $\vec{c} = (3, 2, -5)$.The Cartesian equation of a plane passing through $(x_1, y_1, z_1)$ and parallel to vectors $\vec{b} = (a_1, b_1, c_1)$ and $\vec{c} = (a_2, b_2, c_2)$ is given by the determinant equation:$\begin{vmatrix} x - x_1 & y - y_1 & z - z_1 \ a_1 & b_1 & c_1 \ a_2 & b_2 & c_2 \end{vmatrix} = 0$Substituting the given values:$\begin{vmatrix} x - 3 & y + 2 & z + 1 \ 1 & -2 & 4 \ 3 & 2 & -5 \end{vmatrix} = 0$Expanding the determinant along the first row:$(x - 3)((-2)(-5) - (4)(2)) - (y + 2)((1)(-5) - (4)(3)) + (z + 1)((1)(2) - (-2)(3)) = 0$$(x - 3)(10 - 8) - (y + 2)(-5 - 12) + (z + 1)(2 + 6) = 0$$2(x - 3) + 17(y + 2) + 8(z + 1) = 0$$2x - 6 + 17y + 34 + 8z + 8 = 0$$2x + 17y + 8z + 36 = 0$

The Cartesian equation of the plane passing through the point $(3, -2, -1)$ and parallel to the vectors $\vec{b} = \hat{i} - 2\hat{j} + 4\hat{k}$ and $\vec{c} = 3\hat{i} + 2\hat{j} - 5\hat{k}$ is:

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