Find the vector equation of a plane which passes through the point of intersection of the planes \[ \vec{r} \cdot (\hat{i} + \hat{j} + \hat{k}) = 6 \, \text{and} \, \vec{r} \cdot (2\hat{i} + 3\hat{j} + 4\hat{k}) = -5 \] and the point $(1,1,1)$.
A relation \( R = \{(x, y) : \text{Number of pages in} \, x \text{ and } y \text{ are equal} \} \) is defined on the set \( A \) of all books in a college library. Prove that \( R \) is an equivalence relation.
Find the equation of the plane which passes through the intersecting point of the planes \[ \vec{r} \cdot (\hat{i} + \hat{j} + \hat{k}) = 6 \, \text{and} \, \vec{r} \cdot (2\hat{i} + 3\hat{j} + 4\hat{k}) = -5, \] and the point \( (1, 1, 1) \).