Lesson 3 of 10 · 11 min

The eight octants

NCERT §11.2, Table 11.1

The drone hovers at (3, 4, 12). A kite flies at (−6, 2, 8). A stone lies on the floor of the tank at (−2, 3, −6). Each is in a different region of space. How are those regions named?

The lesson in notes

In short

The three coordinate planes split the whole of space into eight parts called octants, just as the two axes split the plane into four quadrants.

Each is named by the three half-axes that bound it and numbered I to VIII: I is XOYZ, II is X′OYZ, III is X′OY′Z, IV is XOY′Z, V is XOYZ′, VI is X′OYZ′, VII is X′OY′Z′ and VIII is XOY′Z′.

Octants I to IV lie above the XY-plane (z positive) and V to VIII below it (z negative). Each lower octant sits directly under the upper one with the same number minus four: V under I, VI under II, VII under III, VIII under IV.

Signs of (x, y, z): I (+, +, +), II (−, +, +), III (−, −, +), IV (+, −, +), V (+, +, −), VI (−, +, −), VII (−, −, −), VIII (+, −, −).

So the signs alone name the octant: (−3, 1, 2) is in octant II and (−3, 1, −2) is in octant VI. Likewise (2, −5, 4) is in IV and (−1, −1, −1) is in VII.

A point with a zero coordinate lies on a coordinate plane and belongs to no octant.