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 planes given by the equations \( \vec{r} \cdot (\hat{i} + \hat{j} + \hat{k}) = 6 \) and \( \vec{r} \cdot (2\hat{i} + 3\hat{j} + 4\hat{k}) = -5 \), and also passing through the point (1, 1, 1), we can follow these steps:

### Step 1: Write the equations of the given planes
The equations of the planes can be expressed as:
1. Plane 1: \( \vec{r} \cdot (\hat{i} + \hat{j} + \hat{k}) = 6 \)
2. Plane 2: \( \vec{r} \cdot (2\hat{i} + 3\hat{j} + 4\hat{k}) = -5 \)

### Step 2: Find the normal vectors of the planes
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