Find the vector equation of the plane passing through the intersection of the planes ` -> rdot( hat i+ hat j+ hat k)=6`and ` -> rdot(2 hat i+3 hat j+4 hat k)=-5`and the point (1, 1, 1).

Explanation

To find the vector equation of the plane passing through the intersection of the given planes and the point (1, 1, 1), we can follow these steps:

### Step 1: Write the equations of the given planes
The equations of the two planes are:
1. \( P_1: \vec{r} \cdot (\hat{i} + \hat{j} + \hat{k}) = 6 \)
2. \( P_2: \vec{r} \cdot (2\hat{i} + 3\hat{j} + 4\hat{k}) = -5 \)

### Step 2: Express the equation of the plane through the intersection
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