In which octant the following points lie :
(i) (2, 1, 4) (ii) (-1,2,4)
(iii) (1,-3, 2) (iv) (1, 5,-6)
(v) (2, -1, -3) (vi) (-2, 3, -5)
(vii) (-4, -1, 3) (viii) (-1, -2, 3)

Explanation

To determine in which octant the given points lie, we need to analyze the signs of their coordinates (x, y, z). The octants in three-dimensional space are defined based on the signs of these coordinates. Here’s a breakdown of the octants:

1. **First Octant**: x > 0, y > 0, z > 0
2. **Second Octant**: x < 0, y > 0, z > 0
3. **Third Octant**: x < 0, y < 0, z > 0
4. **Fourth Octant**: x > 0, y < 0, z > 0
5. **Fifth Octant**: x > 0, y > 0, z < 0
6. **Sixth Octant**: x < 0, y > 0, z < 0
7. **Seventh Octant**: x < 0, y < 0, z < 0
8. **Eighth Octant**: x > 0, y < 0, z < 0

Now, let's analyze each point one by one:

### Step 1: Analyze Point (2, 1, 4)
- x = 2 (positive)
- y = 1 (positive)
- z = 4 (positive)
- **Conclusion**: This point lies in the **First Octant**.

### Step 2: Analyze Point (-1, 2, 4)
- x = -1 (negative)
- y = 2 (positive)
- z = 4 (positive)
- **Conclusion**: This point lies in the **Second Octant**.

### Step 3: Analyze Point (1, -3, 2)
- x = 1 (positive)
- y = -3 (negative)
- z = 2 (positive)
- **Conclusion**: This point lies in the **Fourth Octant**.

### Step 4: Analyze Point (1, 5, -6)
- x = 1 (positive)
- y = 5 (positive)
- z = -6 (negative)
- **Conclusion**: This point lies in the **Fifth Octant**.

### Step 5: Analyze Point (2, -1, -3)
- x = 2 (positive)
- y = -1 (negative)
- z = -3 (negative)
- **Conclusion**: This point lies in the **Eighth Octant**.

### Step 6: Analyze Point (-2, 3, -5)
- x = -2 (negative)
- y = 3 (positive)
- z = -5 (negative)
- **Conclusion**: This point lies in the **Sixth Octant**.

### Step 7: Analyze Point (-4, -1, 3)
- x = -4 (negative)
- y = -1 (negative)
- z = 3 (positive)
- **Conclusion**: This point lies in the **Third Octant**.

### Step 8: Analyze Point (-1, -2, 3)
- x = -1 (negative)
- y = -2 (negative)
- z = 3 (positive)
- **Conclusion**: This point lies in the **Seventh Octant**.

### Summary of Results:
1. (2, 1, 4) - First Octant
2. (-1, 2, 4) - Second Octant
3. (1, -3, 2) - Fourth Octant
4. (1, 5, -6) - Fifth Octant
5. (2, -1, -3) - Eighth Octant
6. (-2, 3, -5) - Sixth Octant
7. (-4, -1, 3) - Third Octant
8. (-1, -2, 3) - Seventh Octant