Find whether the lines :
`vec(r) = (hat(i) - hat(j) - hat(k)) + lambda(2 hat(i) + hat(j))`
`vecr=(2hati-hatj)+mu(hati+hatj-hatk)`
intersect or not. If intersecting, Find their point of intersection.

Answer

To determine whether the given lines intersect and to find the point of intersection if they do, we can follow these steps:

### Step 1: Write down the equations of the lines in vector form.

The first line is given by:
\[ \vec{r_1} = (\hat{i} - \hat{j} - \hat{k}) + \lambda(2\hat{i} + \hat{j}) \]
This can be rewritten as:
\[ \vec{r_1} = (1 + 2\lambda)\hat{i} + (-1 + \lambda)\hat{j} - \hat{k} \]

The second line is given by:
\[ \vec{r_2} = (2\hat{i} - \hat{j}) + \mu(\hat{i} + \hat{j} - \hat{k}) \]
This can be rewritten as:
\[ \vec{r_2} = (2 + \mu)\hat{i} + (-1 + \mu)\hat{j} - \mu\hat{k} \]

### Step 2: Set the components of the two lines equal to each other.

We need to equate the components of \(\vec{r_1}\) and \(\vec{r_2}\):
1. For the \(\hat{i}\) component:
\[ 1 + 2\lambda = 2 + \mu \quad \text{(1)} \]
2. For the \(\hat{j}\) component:
\[ -1 + \lambda = -1 + \mu \quad \text{(2)} \]
3. For the \(\hat{k}\) component:
\[ -1 = -\mu \quad \text{(3)} \]

### Step 3: Solve the equations.

From equation (3):
\[ \mu = 1 \]

Substituting \(\mu = 1\) into equation (2):
\[ -1 + \lambda = -1 + 1 \implies \lambda = 1 \]

Now substituting \(\lambda = 1\) and \(\mu = 1\) into equation (1):
\[ 1 + 2(1) = 2 + 1 \implies 3 = 3 \]
This is consistent, indicating that the values of \(\lambda\) and \(\mu\) are correct.

### Step 4: Find the point of intersection.

Substituting \(\lambda = 1\) into the first line's equation:
\[ \vec{r_1} = (1 + 2(1))\hat{i} + (-1 + 1)\hat{j} - \hat{k} = 3\hat{i} + 0\hat{j} - 1\hat{k} \]
Thus, the point of intersection is:
\[ (3, 0, -1) \]

### Conclusion

The lines intersect at the point \((3, 0, -1)\).

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