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

Options

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

Explanation

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

### Step 1: Understand the Octants in 3D Space
In three-dimensional Cartesian coordinates, the space is divided into eight octants based on the signs of the coordinates \(x\), \(y\), and \(z\). The octants are defined as follows:

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: Analyze the Coordinates of the Given Point
The given point is \((-2, -3, -4)\). We can analyze the signs of each coordinate:
- \(x = -2\) (negative)
- \(y = -3\) (negative)
- \(z = -4\) (negative)

### Step 3: Determine the Octant Based on the Signs
Since all three coordinates \(x\), \(y\), and \(z\) are negative, we can conclude that the point lies in the seventh octant, where \(x < 0\), \(y < 0\), and \(z < 0\).

### Conclusion
Thus, the point \((-2, -3, -4)\) lies in the **seventh octant**.

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