Q.Name the octants in which the following points lie: $(1, 2, 3)$, $(4, -2, 3)$, $(4, -2, -5)$, $(4, 2, -5)$, $(-4, 2, -5)$, $(-4, 2, 5)$, $(-3, -1, 6)$, $(-2, -4, -7)$.

The key idea is that in 3D coordinate geometry, the three axes divide space into eight octants, identified by the signs of the $x$, $y$, and $z$ coordinates.

Step 1: Recall the sign pattern for each octant:

• Octant I: $(+, +, +)$
• Octant II: $(-, +, +)$
• Octant III: $(-, -, +)$
• Octant IV: $(+, -, +)$
• Octant V: $(+, +, -)$
• Octant VI: $(-, +, -)$
• Octant VII: $(-, -, -)$
• Octant VIII: $(+, -, -)$

Step 2: Match each point’s signs to the correct octant:

• $(1, 2, 3)$ → $(+, +, +)$ → Octant I
• $(4, -2, 3)$ → $(+, -, +)$ → Octant IV
• $(4, -2, -5)$ → $(+, -, -)$ → Octant VIII …

Why Octants Work the Way They Do

In three-dimensional coordinate geometry, the three axes (x, y, z) divide space into eight regions called octants. Unlike the four quadrants in 2D (numbered anticlockwise from the positive x, positive y region), octants are numbered using a specific convention based on the signs of the coordinates.

The standard numbering is:

Notice the pattern: Octants I–IV have positive z, and V–VIII have negative z. Within each z-group, the x and y signs follow the same order as the 2D quadrants (starting from positive x, positive y and going anticlockwise).

Step-by-Step Solution

1. Point $(1, 2, 3)$

All three coordinates are positive: x > 0, y > 0, z > 0. This is the first octant (I).

2. Point $(4, -2, 3)$

x > 0, y < 0, z > 0. Since z > 0, we’re in I–IV. The (x, y) signs are (+, –), which is the 4th quadrant in 2D. So this is octant IV.

3. Point $(4, -2, -5)$

x > 0, y < 0, z < 0. Since z < 0, we’re in V–VIII. The (x, y) signs (+, –) correspond to the 4th quadrant, which in the negative-z group is octant VIII.

4. Point $(4, 2, -5)$ …