The lines `vecr=(hati+hatj)+lamda(hati+hatk)andvecr=(hati+hatj)+mu(-hati+hatj-hatk)` are

Options

A. parallel
B. non-intersecting
C. intersecting
D. None of these

Answer

To determine the relationship between the two lines given by the vector equations:

1. **Line 1:** \(\vec{r} = \hat{i} + \hat{j} + \lambda(\hat{i} + \hat{k})\)
2. **Line 2:** \(\vec{r} = \hat{i} + \hat{j} + \mu(-\hat{i} + \hat{j} - \hat{k})\)

we will analyze their direction vectors and points of intersection.

### Step 1: Identify the direction vectors and points of the lines

For Line 1:
- Point on the line: \(\vec{A} = \hat{i} + \hat{j}\)
- Direction vector: \(\vec{d_1} = \hat{i} + \hat{k}\)

For Line 2:
- Point on the line: \(\vec{B} = \hat{i} + \hat{j}\)
- Direction vector: \(\vec{d_2} = -\hat{i} + \hat{j} - \hat{k}\)

### Step 2: Check if the lines are parallel

Two lines are parallel if their direction vectors are scalar multiples of each other. We need to check if \(\vec{d_1}\) and \(\vec{d_2}\) are parallel.

\[ \vec{d_1} = \hat{i} + \hat{k} \quad \text{and} \quad \vec{d_2} = -\hat{i} + \hat{j} - \hat{k} \]

To check for parallelism, we can set up the equation:

\[ \vec{d_1} = k \cdot \vec{d_2} \]

for some scalar \(k\). This leads to the following system of equations based on the components:

1. \(1 = -k\) (from the \(\hat{i}\) component)
2. \(0 = k\) (from the \(\hat{j}\) component)
3. \(1 = -k\) (from the \(\hat{k}\) component)

From the second equation, \(k\) must be \(0\), which contradicts the first and third equations. Thus, the lines are not parallel.

### Step 3: Check if the lines intersect

To check for intersection, we need to find values of \(\lambda\) and \(\mu\) such that:

\[ \hat{i} + \hat{j} + \lambda(\hat{i} + \hat{k}) = \hat{i} + \hat{j} + \mu(-\hat{i} + \hat{j} - \hat{k}) \]

This simplifies to:

\[ \lambda(\hat{i} + \hat{k}) = \mu(-\hat{i} + \hat{j} - \hat{k}) \]

Equating the components gives us:

1. \(\lambda = -\mu\) (from the \(\hat{i}\) component)
2. \(0 = \mu\) (from the \(\hat{j}\) component)
3. \(\lambda = -\mu\) (from the \(\hat{k}\) component)

From the second equation, we find \(\mu = 0\). Substituting \(\mu = 0\) into the first equation gives \(\lambda = 0\).

### Step 4: Conclusion

When \(\lambda = 0\) and \(\mu = 0\), both lines pass through the point \(\hat{i} + \hat{j}\). Therefore, the lines intersect at this point.

Thus, the final conclusion is that the lines are **intersecting**.