If `a lt 0, b gt 0 and  c lt 0`, then the point `P(a, b, -c)` lies in the octant

Answer

To determine in which octant the point \( P(a, b, -c) \) lies, given the conditions \( a < 0 \), \( b > 0 \), and \( c < 0 \), we can analyze the coordinates of the point step by step.

### Step 1: Identify the coordinates of the point
The point \( P(a, b, -c) \) has the following coordinates:
- \( x \)-coordinate: \( a \)
- \( y \)-coordinate: \( b \)
- \( z \)-coordinate: \( -c \)

### Step 2: Analyze the \( x \)-coordinate
Given that \( a < 0 \), we conclude:
- The \( x \)-coordinate is negative.

### Step 3: Analyze the \( y \)-coordinate
Given that \( b > 0 \), we conclude:
- The \( y \)-coordinate is positive.

### Step 4: Analyze the \( z \)-coordinate
Given that \( c < 0 \), we can deduce:
- Since \( -c \) is the negative of a negative number, \( -c > 0 \).
- Therefore, the \( z \)-coordinate is positive.

### Step 5: Determine the octant
Now we have:
- \( x < 0 \) (negative)
- \( y > 0 \) (positive)
- \( z > 0 \) (positive)

The coordinates \( (x, y, z) \) being \( (negative, positive, positive) \) indicates that the point lies in the second octant of the three-dimensional Cartesian coordinate system.

### Conclusion
Thus, the point \( P(a, b, -c) \) lies in the second octant.

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