Find the vector equation of the plane passing through the intersection of the planes
`vec(r). (2 hati + 2 hatj - 3 hatk) = 7, vecr(r). (2 hati + 5 hatj + 3 hatk ) = 9 `
and through the point (2,1,3).

Answer

To find the vector equation of the plane passing through the intersection of the given planes and the point (2, 1, 3), we can follow these steps:

### Step 1: Write the equations of the given planes
The equations of the planes in vector form are:
1. \( \vec{r} \cdot (2 \hat{i} + 2 \hat{j} - 3 \hat{k}) = 7 \)
2. \( \vec{r} \cdot (2 \hat{i} + 5 \hat{j} + 3 \hat{k}) = 9 \)

### Step 2: Set up the equation of the plane through the intersection
The equation of a plane through the intersection of two planes can be expressed as:
\[ \vec{r} \cdot (2 \hat{i} + 2 \hat{j} - 3 \hat{k}) + \lambda \vec{r} \cdot (2 \hat{i} + 5 \hat{j} + 3 \hat{k}) = 0 \]
This can be rewritten as:
\[ (2 + 2\lambda)x + (2 + 5\lambda)y + (-3 + 3\lambda)z - (7 + 9\lambda) = 0 \]

### Step 3: Substitute the point (2, 1, 3)
We need to ensure that the plane passes through the point (2, 1, 3). Substitute \(x = 2\), \(y = 1\), and \(z = 3\) into the equation:
\[ (2 + 2\lambda)(2) + (2 + 5\lambda)(1) + (-3 + 3\lambda)(3) - (7 + 9\lambda) = 0 \]

### Step 4: Simplify the equation
Expanding the equation:
\[ (4 + 4\lambda) + (2 + 5\lambda) + (-9 + 9\lambda) - (7 + 9\lambda) = 0 \]
Combine like terms:
\[ 4 + 2 - 9 - 7 + (4\lambda + 5\lambda + 9\lambda - 9\lambda) = 0 \]
This simplifies to:
\[ -10 + 9\lambda = 0 \]

### Step 5: Solve for \(\lambda\)
From the equation:
\[ 9\lambda = 10 \implies \lambda = \frac{10}{9} \]

### Step 6: Substitute \(\lambda\) back into the plane equation
Now substitute \(\lambda = \frac{10}{9}\) back into the plane equation:
\[ (2 + 2 \cdot \frac{10}{9})x + (2 + 5 \cdot \frac{10}{9})y + (-3 + 3 \cdot \frac{10}{9})z - (7 + 9 \cdot \frac{10}{9}) = 0 \]
This gives:
\[ \left(\frac{38}{9}\right)x + \left(\frac{68}{9}\right)y + \left(\frac{3}{9}\right)z - \left(\frac{153}{9}\right) = 0 \]

### Step 7: Multiply through by 9 to eliminate the fraction
Multiply the entire equation by 9 to simplify:
\[ 38x + 68y + 3z = 153 \]

### Final Answer
The vector equation of the plane is:
\[ 38x + 68y + 3z = 153 \]