Write the vector equation of the plane passing through the point (a, b, c) and parallel to the plane 𝒓 ⋅ (î + ĵ + k̂) = 2.

Equation of plane: (𝒓 - 𝒂) ⋅ 𝒏 = 0
Given that the plane is parallel to the plane 𝒓 ⋅ (î + ĵ + k̂) = 2, the normal vector 𝒏 is (î + ĵ + k̂).
The point on the plane is (a, b, c), which can be represented as the vector 𝒂 = aî + bĵ + ck̂.
Substituting these values into the equation of the plane:
(𝒓 - (aî + bĵ + ck̂)) ⋅ (î + ĵ + k̂) = 0
Expanding the equation:
𝒓 ⋅ (î + ĵ + k̂) - (aî + bĵ + ck̂) ⋅ (î + ĵ + k̂) = 0
𝒓 ⋅ (î + ĵ + k̂) - (a + b + c) = 0
Therefore, the vector equation of the plane is:
𝒓 ⋅ (î + ĵ + k̂) = a + b + c