If `r=hat(i)+hat(j)+lambda(2hat(i)+hat(j)+4hat(k)) and r*(hat(i)+2hat(j)-hat(k)=3` are equations of a line and a plane respectively, then which of the following is incorrect?

Options

A. line is perpendicular to the plane
B. line lies in the plane
C. line is parallel to the plane but not lie in the plane
D. line cuts the plane obliquely

Answer

To solve the problem, we need to analyze the given equations of the line and the plane, and then determine which option is incorrect based on the relationship between the line and the plane.

### Given:
1. **Line Equation:**
\[ \mathbf{r} = \hat{i} + \hat{j} + \lambda(2\hat{i} + \hat{j} + 4\hat{k}) \]
Here, we can identify:
- Point \( A = \hat{i} + \hat{j} \)
- Direction vector \( \mathbf{b} = 2\hat{i} + \hat{j} + 4\hat{k} \)

2. **Plane Equation:**
\[ \mathbf{r} \cdot (\hat{i} + 2\hat{j} - \hat{k}) = 3 \]
Here, the normal vector \( \mathbf{n} = \hat{i} + 2\hat{j} - \hat{k} \) and \( d = 3 \).

### Step 1: Find the direction vector of the line
The direction vector \( \mathbf{b} \) is:
\[ \mathbf{b} = 2\hat{i} + \hat{j} + 4\hat{k} \]

### Step 2: Find the normal vector of the plane
The normal vector \( \mathbf{n} \) is:
\[ \mathbf{n} = \hat{i} + 2\hat{j} - \hat{k} \]

### Step 3: Check if the line is perpendicular to the plane
To check if the line is perpendicular to the plane, we compute the dot product \( \mathbf{b} \cdot \mathbf{n} \):
\[ \mathbf{b} \cdot \mathbf{n} = (2\hat{i} + \hat{j} + 4\hat{k}) \cdot (\hat{i} + 2\hat{j} - \hat{k}) \]
Calculating this:
\[ = 2 \cdot 1 + 1 \cdot 2 + 4 \cdot (-1) = 2 + 2 - 4 = 0 \]
Since the dot product is zero, the line is perpendicular to the plane.

### Step 4: Check if the line lies in the plane
To check if the line lies in the plane, we substitute the point \( A = \hat{i} + \hat{j} \) into the plane equation:
\[ \mathbf{r} \cdot \mathbf{n} = (\hat{i} + \hat{j}) \cdot (\hat{i} + 2\hat{j} - \hat{k}) \]
Calculating this:
\[ = 1 \cdot 1 + 1 \cdot 2 + 0 = 1 + 2 + 0 = 3 \]
Since this equals \( d = 3 \), the point lies in the plane.

### Step 5: Determine the relationship between the line and the plane
Since the line is perpendicular to the plane, it cannot be parallel to the plane. Therefore, any statement claiming that the line is parallel to the plane is incorrect.

### Conclusion
The incorrect statement among the options provided is that the line is parallel to the plane.