Find the octant in which the following points lie :
(i) (2,3,-5) (ii) (-3,1,4)
(iii) (-1,-1,2) (iv) (1,1,3)
(v) (-5,-4,-1) (vi) (4,-1,2)

Explanation

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

1. **1st Octant**: x > 0, y > 0, z > 0
2. **2nd Octant**: x < 0, y > 0, z > 0
3. **3rd Octant**: x < 0, y < 0, z > 0
4. **4th Octant**: x > 0, y < 0, z > 0
5. **5th Octant**: x > 0, y > 0, z < 0
6. **6th Octant**: x < 0, y > 0, z < 0
7. **7th Octant**: x < 0, y < 0, z < 0
8. **8th Octant**: x > 0, y < 0, z < 0

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

### Step 1: Analyze Point (2, 3, -5)
- x = 2 (positive)
- y = 3 (positive)
- z = -5 (negative)

Since x > 0, y > 0, and z < 0, this point lies in the **5th Octant**.

### Step 2: Analyze Point (-3, 1, 4)
- x = -3 (negative)
- y = 1 (positive)
- z = 4 (positive)

Since x < 0, y > 0, and z > 0, this point lies in the **2nd Octant**.

### Step 3: Analyze Point (-1, -1, 2)
- x = -1 (negative)
- y = -1 (negative)
- z = 2 (positive)

Since x < 0, y < 0, and z > 0, this point lies in the **3rd Octant**.

### Step 4: Analyze Point (1, 1, 3)
- x = 1 (positive)
- y = 1 (positive)
- z = 3 (positive)

Since x > 0, y > 0, and z > 0, this point lies in the **1st Octant**.

### Step 5: Analyze Point (-5, -4, -1)
- x = -5 (negative)
- y = -4 (negative)
- z = -1 (negative)

Since x < 0, y < 0, and z < 0, this point lies in the **7th Octant**.

### Step 6: Analyze Point (4, -1, 2)
- x = 4 (positive)
- y = -1 (negative)
- z = 2 (positive)

Since x > 0, y < 0, and z > 0, this point lies in the **4th Octant**.

### Summary of Results:
1. (2, 3, -5) → 5th Octant
2. (-3, 1, 4) → 2nd Octant
3. (-1, -1, 2) → 3rd Octant
4. (1, 1, 3) → 1st Octant
5. (-5, -4, -1) → 7th Octant
6. (4, -1, 2) → 4th Octant