Find out whether the following pairs of lines are parallel, non parallel, & intersecting, or non-parallel & non-intersecting.
`vec(r_(1)) = hat(i) - hat(j) + 3hat(k) + lambda(hat(i) - hat(j)+hat(k))`
`vec(r_(2)) = hat(i) - hat(j) + 3hat(k) + lambda(hat(i) - hat(j) + hat(k))`
`vec(r_(2)) = 2hat(i) + 4hat(j) + 6hat(k) + mu (2hat(i) + hat(j) + 3hat(k))`

Answer

To determine whether the given lines are parallel, non-parallel and intersecting, or non-parallel and non-intersecting, we will analyze the equations of the lines step-by-step.

### Step 1: Identify the equations of the lines

The equations of the lines are given as:

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

2. **Line 2**:
\[ \vec{r_2} = 2\hat{i} + 4\hat{j} + 6\hat{k} + \mu(2\hat{i} + \hat{j} + 3\hat{k}) \]

### Step 2: Extract direction ratios and points

For **Line 1**:
- Point: \( (1, -1, 3) \)
- Direction ratios: \( (1, -1, 1) \)

For **Line 2**:
- Point: \( (2, 4, 6) \)
- Direction ratios: \( (2, 1, 3) \)

### Step 3: Check for parallelism

To check if the lines are parallel, we need to see if the direction ratios are proportional. We check if:

\[ \frac{1}{2} = \frac{-1}{1} = \frac{1}{3} \]

Calculating these ratios:
- \( \frac{1}{2} \) is not equal to \( -1 \)
- \( \frac{1}{2} \) is not equal to \( \frac{1}{3} \)

Since the ratios are not equal, the lines are **not parallel**.

### Step 4: Check for intersection

To check if the lines intersect, we can use the condition that if the lines intersect, the following must hold true:

\[ \frac{x_1 - x_2}{a_1} = \frac{y_1 - y_2}{b_1} = \frac{z_1 - z_2}{c_1} \]

Where \( (x_1, y_1, z_1) \) and \( (x_2, y_2, z_2) \) are the points on the two lines, and \( (a_1, b_1, c_1) \) and \( (a_2, b_2, c_2) \) are the direction ratios.

Using the points:
- Point from Line 1: \( (1, -1, 3) \)
- Point from Line 2: \( (2, 4, 6) \)

Calculating:
- \( x_1 - x_2 = 1 - 2 = -1 \)
- \( y_1 - y_2 = -1 - 4 = -5 \)
- \( z_1 - z_2 = 3 - 6 = -3 \)

Now we check the ratios:
\[ \frac{-1}{1} = -1, \quad \frac{-5}{-1} = 5, \quad \frac{-3}{1} = -3 \]

Since the ratios \( -1 \), \( 5 \), and \( -3 \) are not equal, the lines do not intersect.

### Conclusion

The given lines are **non-parallel and non-intersecting**.

---