Find the angle between the line:
`vec(r) = (hat(i) - hat(j) + hat(k) ) + lambda (2 hat(i) - hat(j) + 3 hat(k))` and the plane `vec(r). (2 hat(i) + hat(j) - hat(k) ) = 4. `
Also, find the whether the line is parallel to the plane or not.

Answer

To find the angle between the given line and the plane, we will follow these steps:

### Step 1: Identify the Direction Ratios of the Line and the Plane
The line is given in the vector form:
\[ \vec{r} = (\hat{i} - \hat{j} + \hat{k}) + \lambda(2\hat{i} - \hat{j} + 3\hat{k}) \]
From this, we can extract the direction ratios of the line (denoted as vector \( \vec{b} \)):
- The direction ratios of the line are \( 2, -1, 3 \).

The equation of the plane is given as:
\[ \vec{r} \cdot (2\hat{i} + \hat{j} - \hat{k}) = 4 \]
The normal vector (denoted as vector \( \vec{n} \)) to the plane can be directly obtained from the coefficients of \( \hat{i}, \hat{j}, \hat{k} \):
- The direction ratios of the plane are \( 2, 1, -1 \).

### Step 2: Use the Formula for the Angle Between a Line and a Plane
The angle \( \theta \) between the line and the plane can be found using the formula:
\[ \sin \theta = \frac{\vec{n} \cdot \vec{b}}{|\vec{n}| |\vec{b}|} \]
Where:
- \( \vec{n} = 2\hat{i} + 1\hat{j} - 1\hat{k} \)
- \( \vec{b} = 2\hat{i} - 1\hat{j} + 3\hat{k} \)

### Step 3: Calculate the Dot Product \( \vec{n} \cdot \vec{b} \)
Calculating the dot product:
\[ \vec{n} \cdot \vec{b} = (2)(2) + (1)(-1) + (-1)(3) = 4 - 1 - 3 = 0 \]

### Step 4: Calculate the Magnitudes of \( \vec{n} \) and \( \vec{b} \)
Now, we calculate the magnitudes:
\[ |\vec{n}| = \sqrt{(2^2) + (1^2) + (-1^2)} = \sqrt{4 + 1 + 1} = \sqrt{6} \]
\[ |\vec{b}| = \sqrt{(2^2) + (-1^2) + (3^2)} = \sqrt{4 + 1 + 9} = \sqrt{14} \]

### Step 5: Substitute into the Formula
Substituting these values into the sine formula:
\[ \sin \theta = \frac{0}{\sqrt{6} \cdot \sqrt{14}} = 0 \]

### Step 6: Determine the Angle \( \theta \)
Since \( \sin \theta = 0 \), this implies:
\[ \theta = 0^\circ \text{ or } \theta = 180^\circ \]
This means the line is either parallel to the plane or in the opposite direction.

### Step 7: Conclusion on Parallelism
Since the sine of the angle is zero, we conclude that the line is parallel to the plane.

### Final Answer
- The angle between the line and the plane is \( 0^\circ \) (or \( 180^\circ \)).
- The line is parallel to the plane.

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