Question:

First, check if the planes are parallel by comparing normal vectors. For $P_1: \vec{n_1} = 2\hat{i} + \hat{j} + \hat{k}$ and $P_2: \vec{n_2} = 2\hat{i} - \hat{j} + \hat{k}$. Since the planes are not parallel (normal vectors are different), we find the line of intersection. Setting up the system from both plane equations and solving, the line of intersection is parallel to $\vec{n_1} × \vec{n_2}$. Computing the distance from a point on $P_1$ (such as $(1, 2, 3)$) to $P_2$ using the point-to-plane distance formula: $d = \frac{|ax_0 + by_0 + cz_0 + d|}{\sqrt{a^2 + b^2 + c^2}}$, which yields $-5/\sqrt{11}$.