Find the point of intersection of the line :
`vec(r) = (hat(i) + 2 hat(j) + 3 hat(k) ) + lambda (2 hat(i) + hat(j) + 2 hat(k))` and the plane `vec(r). (2 hat(i) - 6 hat(j) + 3 hat(k) ) ` + 5 = 0.

Answer

To find the point of intersection of the given line and plane, we can follow these steps:

### Step 1: Write the equation of the line in parametric form
The equation of the line is given as:
\[ \vec{r} = (\hat{i} + 2\hat{j} + 3\hat{k}) + \lambda(2\hat{i} + \hat{j} + 2\hat{k}) \]
From this, we can extract the parametric equations for \(x\), \(y\), and \(z\):
- \(x = 1 + 2\lambda\)
- \(y = 2 + \lambda\)
- \(z = 3 + 2\lambda\)

### Step 2: Write the equation of the plane in Cartesian form
The equation of the plane is given as:
\[ \vec{r} \cdot (2\hat{i} - 6\hat{j} + 3\hat{k}) + 5 = 0 \]
This can be expanded to:
\[ 2x - 6y + 3z + 5 = 0 \]

### Step 3: Substitute the parametric equations into the plane's equation
Now, we substitute the values of \(x\), \(y\), and \(z\) from the line's equations into the plane's equation:
\[ 2(1 + 2\lambda) - 6(2 + \lambda) + 3(3 + 2\lambda) + 5 = 0 \]

### Step 4: Simplify the equation
Expanding this gives:
\[ 2 + 4\lambda - 12 - 6\lambda + 9 + 6\lambda + 5 = 0 \]
Combining like terms:
\[ (2 - 12 + 9 + 5) + (4\lambda - 6\lambda + 6\lambda) = 0 \]
This simplifies to:
\[ 4 = 0 \]
This means that the terms involving \(\lambda\) cancel out, and we are left with a constant.

### Step 5: Solve for \(\lambda\)
Since the equation simplifies to \(4 = 0\), it indicates that there is no specific \(\lambda\) that satisfies this equation, meaning the line lies entirely in the plane.

### Step 6: Find the intersection point
To find a specific point on the line, we can choose any value for \(\lambda\). Let's choose \(\lambda = 0\):
- \(x = 1 + 2(0) = 1\)
- \(y = 2 + (0) = 2\)
- \(z = 3 + 2(0) = 3\)

Thus, one point on the line is \((1, 2, 3)\).

### Step 7: Conclusion
Since the line lies in the plane, every point on the line is an intersection point. The specific point we calculated is:
\[ \text{Point of intersection: } (1, 2, 3) \]