Statement 1 : Lines `vecr=hati+hatj-hatk+lamda(3hati-hatj) and vecr=4hati-hatk+ mu (2hati+ 3hatk)` intersect.
Statement 2 : If `vecbxxvecd=vec0`, then lines `vecr=veca+lamdavecb and vecr= vecc+lamdavecd` do not 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 given question, we will analyze both statements step by step.

### Statement 1:
We have two lines represented in vector form:

1. \( \vec{r_1} = \hat{i} + \hat{j} - \hat{k} + \lambda(3\hat{i} - \hat{j}) \)
2. \( \vec{r_2} = 4\hat{i} - \hat{k} + \mu(2\hat{i} + 3\hat{k}) \)

To check if these lines intersect, we need to equate \( \vec{r_1} \) and \( \vec{r_2} \):

\[ \vec{r_1} = \hat{i} + \hat{j} - \hat{k} + \lambda(3\hat{i} - \hat{j}) = (1 + 3\lambda)\hat{i} + (1 - \lambda)\hat{j} - \hat{k} \]

\[ \vec{r_2} = 4\hat{i} - \hat{k} + \mu(2\hat{i} + 3\hat{k}) = (4 + 2\mu)\hat{i} + 0\hat{j} + (-1 + 3\mu)\hat{k} \]

Now, we equate the coefficients of \( \hat{i} \), \( \hat{j} \), and \( \hat{k} \):

1. For \( \hat{i} \):
\[ 1 + 3\lambda = 4 + 2\mu \quad \text{(1)} \]

2. For \( \hat{j} \):
\[ 1 - \lambda = 0 \quad \Rightarrow \quad \lambda = 1 \quad \text{(2)} \]

3. For \( \hat{k} \):
\[ -1 + 3\mu = -1 \quad \Rightarrow \quad 3\mu = 0 \quad \Rightarrow \quad \mu = 0 \quad \text{(3)} \]

Now substituting \( \lambda = 1 \) and \( \mu = 0 \) into equation (1):

\[ 1 + 3(1) = 4 + 2(0) \quad \Rightarrow \quad 4 = 4 \]

This confirms that both conditions are satisfied, thus the lines intersect.

### Conclusion for Statement 1:
Both lines intersect at \( \lambda = 1 \) and \( \mu = 0 \). Therefore, Statement 1 is **correct**.

---

### Statement 2:
The second statement states that if \( \vec{b} \times \vec{d} = \vec{0} \), then the lines represented by:

1. \( \vec{r} = \vec{a} + \lambda \vec{b} \)
2. \( \vec{r} = \vec{c} + \lambda \vec{d} \)

do not intersect.

Given that \( \vec{b} \times \vec{d} = \vec{0} \), this implies that \( \vec{b} \) and \( \vec{d} \) are parallel. When two lines are parallel, they can either be the same line or distinct lines that never meet.

To determine if they intersect, we can analyze the conditions for intersection. If the lines are parallel and not coincident, they will not intersect.

### Conclusion for Statement 2:
Since \( \vec{b} \) is parallel to \( \vec{d} \), the lines do not intersect unless they are the same line. Therefore, Statement 2 is also **correct**.

---

### Final Conclusion:
Both statements are correct, but Statement 2 does not provide a correct explanation for Statement 1.

---