the point `( -2,-3,-4)` lies in the

Options

A. first octant
B. seventh octant
C. second octant
D. eight octant

Explanation

To determine which octant the point \((-2, -3, -4)\) lies in, we can follow these steps:

### Step 1: Understand the Octants in 3D Space
In three-dimensional space, octants are defined based on the signs of the coordinates \(x\), \(y\), and \(z\). There are a total of eight 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**: \(x > 0\), \(y < 0\), \(z < 0\)

### Step 2: Identify the Signs of the Coordinates
For the point \((-2, -3, -4)\):
- \(x = -2\) (negative)
- \(y = -3\) (negative)
- \(z = -4\) (negative)

### Step 3: Determine the Octant
Since all three coordinates \(x\), \(y\), and \(z\) are negative, we can refer to the list of octants defined earlier. The only octant where all three coordinates are negative is the **Seventh Octant**.

### Conclusion
The point \((-2, -3, -4)\) lies in the **Seventh Octant**.

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