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) + lambda(hat(i) + 3hat(j) + 4hat(k))`
`vec(r_(2)) = 2hat(i) + 3hat(j) + mu(4hat(i) - hat(j) + hat(k))`

Answer

To determine whether the given lines are parallel, non-parallel and intersecting, or non-parallel and non-intersecting, we will analyze the direction vectors and points of the lines.

### Given Lines:
1. **Line 1**: \(\vec{r_1} = \hat{i} - \hat{j} + \lambda(\hat{i} + 3\hat{j} + 4\hat{k})\)
2. **Line 2**: \(\vec{r_2} = 2\hat{i} + 3\hat{j} + \mu(4\hat{i} - \hat{j} + \hat{k})\)

### Step 1: Identify Direction Vectors
From the equations of the lines, we can identify the direction vectors:

- For Line 1, the direction vector \(\vec{d_1} = \hat{i} + 3\hat{j} + 4\hat{k}\).
- For Line 2, the direction vector \(\vec{d_2} = 4\hat{i} - \hat{j} + \hat{k}\).

### Step 2: Check for Parallelism
Two lines are parallel if their direction vectors are scalar multiples of each other. We can check this by comparing the ratios of the components of the direction vectors:

\[ \frac{1}{4}, \quad \frac{3}{-1}, \quad \frac{4}{1} \]

Calculating these ratios:
- \(\frac{1}{4} \neq \frac{3}{-1} \neq \frac{4}{1}\)

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

### Step 3: Check for Intersection
To check if the lines intersect, we can set the parametric equations equal to each other:

From Line 1:
- \(x = 1 + \lambda\)
- \(y = -1 + 3\lambda\)
- \(z = 4 + 4\lambda\)

From Line 2:
- \(x = 2 + 4\mu\)
- \(y = 3 - \mu\)
- \(z = \mu\)

Setting the equations equal gives us a system of equations:
1. \(1 + \lambda = 2 + 4\mu\)
2. \(-1 + 3\lambda = 3 - \mu\)
3. \(4 + 4\lambda = \mu\)

### Step 4: Solve the System of Equations
From equation (3):
\[ \mu = 4 + 4\lambda \]

Substituting \(\mu\) into equation (1):
\[ 1 + \lambda = 2 + 4(4 + 4\lambda) \]
\[ 1 + \lambda = 2 + 16 + 16\lambda \]
\[ 1 + \lambda = 18 + 16\lambda \]
\[ 1 - 18 = 16\lambda - \lambda \]
\[ -17 = 15\lambda \implies \lambda = -\frac{17}{15} \]

Now substituting \(\lambda\) back into \(\mu\):
\[ \mu = 4 + 4\left(-\frac{17}{15}\right) = 4 - \frac{68}{15} = \frac{60}{15} - \frac{68}{15} = -\frac{8}{15} \]

### Step 5: Verify Intersection
Now we can check if these values of \(\lambda\) and \(\mu\) satisfy the second equation:
Substituting \(\lambda = -\frac{17}{15}\) into equation (2):
\[ -1 + 3\left(-\frac{17}{15}\right) = 3 - \left(-\frac{8}{15}\right) \]
Calculating both sides:
Left side:
\[ -1 - \frac{51}{15} = -\frac{15}{15} - \frac{51}{15} = -\frac{66}{15} \]
Right side:
\[ 3 + \frac{8}{15} = \frac{45}{15} + \frac{8}{15} = \frac{53}{15} \]

Since \(-\frac{66}{15} \neq \frac{53}{15}\), the lines do not intersect.

### Conclusion
The lines are non-parallel and non-intersecting.