What is the equation of the plane P?

Explanation

The family of planes passing through the intersection is (6x+4y-5z-2) + \lambda(x-2y+3z) = 0. The normal vector is \vec{n_P} = \langle 6+\lambda, 4-2\lambda, -5+3\lambda \rangle. Since P is parallel to the line with direction ratios \langle 1, 3, 2 \rangle, their dot product is zero: 1(6+\lambda) + 3(4-2\lambda) + 2(-5+3\lambda) = 0. This gives 6 + \lambda + 12 - 6\lambda - 10 + 6\lambda = 0 \implies 8 + \lambda = 0 \implies \lambda = -8. Substituting \lambda = -8 gives -2x + 20y - 29z - 2 = 0, or 2x - 20y + 29z + 2 = 0.

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