(i) Find the equations of the plane passing through (a,b,c) and parallel to the plane `vec(r) . (hati + hatj + hatk) = 2 .`
(ii) Find the vector equation of the plane through the point `hati + hatj + hatk ` and parallel to the plane `vec(r). (2 hati - hatj + 2 hatk) = 5 `.

Answer

To solve the given questions step by step, we will derive the equations of the planes as specified.

### (i) Find the equations of the plane passing through (a, b, c) and parallel to the plane \(\vec{r} \cdot (\hat{i} + \hat{j} + \hat{k}) = 2\).

1. **Identify the normal vector of the given plane**:
The equation of the plane is given as \(\vec{r} \cdot (\hat{i} + \hat{j} + \hat{k}) = 2\). The normal vector \(\vec{n}\) to this plane is \((1, 1, 1)\) because it is the coefficient of \(\hat{i}, \hat{j}, \hat{k}\).

2. **Use the point-normal form of the plane**:
The point-normal form of a plane is given by the equation:
\[ \vec{r} \cdot \vec{n} = \vec{n} \cdot \vec{a} \]
where \(\vec{a}\) is a point on the plane. Here, \(\vec{a} = (a, b, c)\) and \(\vec{n} = (1, 1, 1)\).

3. **Calculate \(\vec{n} \cdot \vec{a}\)**:
\[ \vec{n} \cdot \vec{a} = (1, 1, 1) \cdot (a, b, c) = a + b + c \]

4. **Write the equation of the required plane**:
Substituting back into the point-normal form:
\[ \vec{r} \cdot (1, 1, 1) = a + b + c \]
This can be expressed as:
\[ x + y + z = a + b + c \]