Q.In which octant does the point $(-4, 2, 5)$ lie?

3D Coordinate Octants

Stand at the corner of a room where two walls meet the floor. That corner is the origin $O$, and the three edges meeting there are the three coordinate axes:

• the edge where the floor meets one wall is the x-axis,
• the edge where the floor meets the other wall is the y-axis,
• the vertical edge where the two walls meet is the z-axis.

These three axes are mutually perpendicular. Taken in pairs they determine three coordinate planes — the $xy$-plane, the $yz$-plane and the $zx$-plane — and these planes slice all of space into eight regions. Each region is called an octant (from octo, meaning eight).

Why exactly eight

In a plane, the two axes make $4$ quadrants. Adding a third axis doubles this: each of the three coordinate planes splits space into two halves, so there are $2 \times 2 \times 2 = 2^3 = 8$ regions. Equivalently, an octant is fixed by choosing a sign — positive or negative — for each of $x$, $y$ and $z$, and there are $2^3 = 8$ such sign-triples $(\pm, \pm, \pm)$.

The eight octants and their signs

The standard NCERT labelling of the signs is:

Notice the pattern: octants I–IV all have $z > 0$ (above the $xy$-plane) and V–VIII all have $z < 0$ (below it).

How to read a point's octant

Just look at the signs of its coordinates. For example, $(-3, 1, 2)$ has $x < 0$, $y > 0$, $z > 0$, which is the sign pattern of octant II. The point $(-3, 1, -2)$ has $x < 0$, $y > 0$, $z < 0$, placing it in octant VI. …