Statement 1 : Lines `vecr= hati-hatj+ lamda (hati+hatj-hatk) and vecr= 2hati-hatj+ mu (hati+hatj-hatk)` do not intersect.
Statement 2 : Skew lines never intersect.

Options

A. Both the statements are true, and Statement 2 is the correct explanation for Statement 1.
B. Both the Statements are true, but Statement 2 is not the correct explanation for Statement 1.
C. Statement 1 is true and Statement 2 is false.
D. Statement 1 is false and Statement 2 is true.

Explanation

To solve the problem, we need to analyze the two lines given in the question and determine whether they intersect or not.

### Step-by-Step Solution:

1. **Identify the lines:**
The first line is given by:
\[ \vec{r_1} = \hat{i} - \hat{j} + \lambda (\hat{i} + \hat{j} - \hat{k}) \]
The second line is given by:
\[ \vec{r_2} = 2\hat{i} - \hat{j} + \mu (\hat{i} + \hat{j} - \hat{k}) \]

2. **Extract points and direction vectors:**
For the first line, we can identify:
- Point \( P_1 = (1, -1, 0) \) (from \(\hat{i} - \hat{j}\))
- Direction vector \( \vec{d_1} = (1, 1, -1) \) (from \(\lambda (\hat{i} + \hat{j} - \hat{k})\))

For the second line, we have:
- Point \( P_2 = (2, -1, 0) \) (from \(2\hat{i} - \hat{j}\))
- Direction vector \( \vec{d_2} = (1, 1, -1) \) (from \(\mu (\hat{i} + \hat{j} - \hat{k})\))

3. **Check if the lines are parallel:**
The direction vectors of both lines are the same:
\[ \vec{d_1} = \vec{d_2} = (1, 1, -1) \]
Since the direction vectors are identical, the lines are parallel.

4. **Check if the lines intersect:**
To check for intersection, we need to see if there exists values of \(\lambda\) and \(\mu\) such that:
\[ P_1 + \lambda \vec{d_1} = P_2 + \mu \vec{d_2} \]
This translates to the following equations:
\[ (1 + \lambda) = (2 + \mu) \quad \text{(1)} \]
\[ (-1 + \lambda) = (-1 + \mu) \quad \text{(2)} \]
\[ (0 - \lambda) = (0 - \mu) \quad \text{(3)} \]

From equation (2):
\[ \lambda = \mu \]

Substituting \(\lambda = \mu\) into equation (1):
\[ 1 + \lambda = 2 + \lambda \implies 1 = 2 \quad \text{(not possible)} \]

Therefore, there are no values of \(\lambda\) and \(\mu\) that satisfy all three equations simultaneously, confirming that the lines do not intersect.

5. **Conclusion:**
- Statement 1 is true: The lines do not intersect.
- Statement 2 is also true: Skew lines never intersect, but in this case, since the lines are parallel, they are not skew lines.

### Final Answer:
- Statement 1 is true, and Statement 2 is also true, but Statement 2 does not correctly explain Statement 1.