Locate the following points: (i) (1,-1,3), (ii) (-1,2,4), (iii) (-2,-4,-7), (iv) (-4,2,-5).

NCERT Exemplar · Q1

Q. Locate the following points:

(i)

(1,-1,3)

(ii)

(-1,2,4)

(iii)

(-2,-4,-7)

(iv)

(-4,2,-5)

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Concept understanding — 3D Coordinate Octants

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

(x, y, z)

The standard NCERT labelling of the signs is:

Octant	Sign of $x$	Sign of $y$	Sign of $z$	Example point
I	$+$	$+$	$+$	$(1, 2, 3)$
II	$-$	$+$	$+$	$(-1, 2, 3)$
III	$-$	$-$	$+$	$(-1, -2, 3)$
IV	$+$	$-$	$+$	$(1, -2, 3)$
V	$+$	$+$	$-$	$(1, 2, -3)$
VI	$-$	$+$	$-$	$(-1, 2, -3)$
VII	$-$	$-$	$-$	$(-1, -2, -3)$
VIII	$+$	$-$	$-$	$(1, -2, -3)$

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.

Watch out

Do not confuse octants (3D, eight regions) with quadrants (2D, four regions). A point with a zero coordinate, such as $(1, -2, 0)$ , lies on a coordinate plane — here the $xy$ -plane — and so belongs to none of the eight octants. Points on the axes and on the coordinate planes are boundaries.

Why it matters

Knowing octants helps you visualise where a point sits relative to the origin, keep track of sign conventions when drawing 3D figures, and quickly sketch surfaces. It is a first step before the distance formula and later work with vectors and lines in space.

This topic sits in the NCERT Class 11 Mathematics chapter Introduction to Three Dimensional Geometry, matching searches such as "octants in 3D geometry class 11 maths" and "three dimensional geometry class 11 important questions". The same sign-based classification of space recurs in JEE Main coordinate geometry and vector algebra, where naming a point's octant at a glance saves valuable exam time.

The octant of a point in 3D is determined by the signs of its $x$ , $y$ , and $z$ coordinates. The eight octants are numbered I through VIII, with Octant I being $(+,+,+)$ and the rest following a specific sign pattern. The points are located in: (i) Octant IV,

(ii) Octant II,

(iii) Octant VII,

(iv) Octant VI.

The Concept: 3D Coordinate Octants

In three-dimensional coordinate geometry, the three axes — $x$ , $y$ , and $z$ — divide space into eight regions called octants. Think of it like the eight rooms formed by three mutually perpendicular walls meeting at a corner.

The key is the sign pattern of the coordinates $(x, y, z)$ . In the first octant (Octant I), all three coordinates are positive: $(+,+,+)$ . From there, the numbering follows a convention that depends on which signs are negative.

Here is the standard sign convention for all eight octants:

Octant	Sign of $x$	Sign of $y$	Sign of $z$
I	$+$	$+$	$+$
II	$-$	$+$	$+$
III	$-$	$-$	$+$
IV	$+$	$-$	$+$
V	$+$	$+$	$-$
VI	$-$	$+$	$-$
VII	$-$	$-$	$-$
VIII	$+$	$-$	$-$

Tip

Notice the pattern: Octants I through IV have $z$ positive (the "upper" half of space), and Octants V through VIII have $z$ negative (the "lower" half). Within each half, the $x$ and $y$ signs cycle through $(+,+)$ , $(-,+)$ , $(-,-)$ , $(+,-)$ — exactly like the four quadrants of the $xy$ -plane.

Now let's locate each point.

Point (i): $(1, -1, 3)$
- $x = 1$ → positive
- $y = -1$ → negative
- $z = 3$ → positive
Since $z$ is positive, we are in the upper half (Octants I–IV). The sign pattern for $x$ and $y$ is $(+, -)$ . Looking at the table, this matches Octant IV.
Point (ii): $(-1, 2, 4)$
- $x = -1$ → negative
- $y = 2$ → positive
- $z = 4$ → positive
Again $z$ is positive. The pattern $(-, +)$ corresponds to Octant II.
Point (iii): $(-2, -4, -7)$
- $x = -2$ → negative
- $y = -4$ → negative
- $z = -7$ → negative
Here $z$ is negative, so we are in the lower half (Octants V–VIII). The pattern $(-, -)$ for $x$ and $y$ gives Octant VII.
Point (iv): $(-4, 2, -5)$
- $x = -4$ → negative
- $y = 2$ → positive
- $z = -5$ → negative
$z$ is negative. The pattern $(-, +)$ for $x$ and $y$ gives Octant VI.

Watch out

A common mistake is to confuse Octant II with Octant VI, or Octant IV with Octant VIII. Always check the sign of $z$ first — that tells you which half of space you're in. Then match the $x$ and $y$ signs to the correct quadrant pattern within that half.

✓Final answer

The points are located in: (i) Octant IV, (ii) Octant II, (iii) Octant VII, (iv) Octant VI.

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The Concept: 3D Coordinate Octants

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