Introduction to Three Dimensional Geometry • Topic 1 of 3

Coordinate Axes, Coordinate Planes and Octants

In Class 10 you fixed the position of a point in a plane with two numbers $(x, y)$. To locate a point in space you need a third measurement — how far the point is above or below the plane — so every point now carries an ordered triple $(x, y, z)$.

Take three mutually perpendicular lines meeting at a single point $O$, the origin. These are the coordinate axes: the $x$-axis, the $y$-axis and the $z$-axis. Each axis is a number line, positive on one side of $O$ and negative on the other.

$$\text{A point in space} \;\longleftrightarrow\; \text{an ordered triple } (x,\ y,\ z)$$

The three coordinate planes: Taken two at a time, the axes determine three planes that slice space apart.

PlaneContains axesEquationPoints on it satisfy
$XY$-plane$x$- and $y$-axis$z = 0$third coordinate is $0$
$YZ$-plane$y$- and $z$-axis$x = 0$first coordinate is $0$
$ZX$-plane$z$- and $x$-axis$y = 0$second coordinate is $0$

Reading the coordinates of a point: For a point $P(x, y, z)$, the number $x$ is the perpendicular distance of $P$ from the $YZ$-plane, $y$ from the $ZX$-plane and $z$ from the $XY$-plane — each taken with the appropriate sign. A point lies on an axis when its other two coordinates are zero (e.g. $(5, 0, 0)$ is on the $x$-axis) and on a coordinate plane when one coordinate is zero.

The eight octants: The three coordinate planes cut space into eight compartments called octants. The first octant is where all three coordinates are positive; the others follow the sign pattern below.

Octant$x$$y$$z$Example point
I$+$$+$$+$$(2, 3, 4)$
II$-$$+$$+$$(-2, 3, 4)$
III$-$$-$$+$$(-2, -3, 4)$
IV$+$$-$$+$$(2, -3, 4)$
V$+$$+$$-$$(2, 3, -4)$
VI$-$$+$$-$$(-2, 3, -4)$
VII$-$$-$$-$$(-2, -3, -4)$
VIII$+$$-$$-$$(2, -3, -4)$

Deeper Insight — three dimensions as two dimensions plus a height: The whole framework of 3D geometry is built by bolting one more perpendicular axis onto the familiar 2D plane, and almost every formula you meet later is the 2D version with a single extra term tagged on. Notice the symmetry running through the tables: the first four octants are exactly the four quadrants of the $XY$-plane "lifted" into positive $z$, and octants V–VIII are their mirror images below the $XY$-plane — so the top four all carry $z $ 0>3431$ and the bottom four $z $ 0<3464$. This is why a coordinate being zero is so informative: one zero pins the point to a coordinate plane, two zeros pin it to an axis, and three zeros give the origin itself. Holding this "plane plus height" picture firmly in mind makes the distance and section formulas in the next two topics feel like old friends rather than new rules.

Three dimensional coordinate axes with a labelled point P(x, y, z) 3D Axes and the Point P(x, y, z) x y z O P(x, y, z) xyz The first octant carries all positive coordinates First Octant: all coordinates positive +x +y +z (+, +, +)
Solved Examples
1
Worked Example
Name the octant in which each point lies: $A(3, -2, 5)$, $B(-4, -1, -6)$, $C(-2, 5, 1)$.
Solution
  1. $A(3, -2, 5)$ has signs $(+, -, +)$ — that is octant IV.
  2. $B(-4, -1, -6)$ has signs $(-, -, -)$ — that is octant VII.
  3. $C(-2, 5, 1)$ has signs $(-, +, +)$ — that is octant II.

Answer: $A$ in octant IV, $B$ in octant VII, $C$ in octant II.

2
Worked Example
On which coordinate plane or axis does each point lie: $P(0, 4, -3)$, $Q(7, 0, 0)$, $R(2, -5, 0)$?
Solution
  1. $P(0, 4, -3)$: the $x$-coordinate is $0$, so $P$ lies on the $YZ$-plane.
  2. $Q(7, 0, 0)$: both $y$ and $z$ are $0$, so $Q$ lies on the $x$-axis.
  3. $R(2, -5, 0)$: the $z$-coordinate is $0$, so $R$ lies on the $XY$-plane.

Answer: $P$ on the $YZ$-plane, $Q$ on the $x$-axis, $R$ on the $XY$-plane.

3
Worked Example
Find the coordinates of the feet of the perpendiculars drawn from $P(3, -4, 5)$ to each of the three coordinate planes.
Solution
  1. Foot on the $XY$-plane: set $z = 0$, keep $x, y$ — gives $(3, -4, 0)$.
  2. Foot on the $YZ$-plane: set $x = 0$ — gives $(0, -4, 5)$.
  3. Foot on the $ZX$-plane: set $y = 0$ — gives $(3, 0, 5)$.

Answer: $(3, -4, 0)$ on $XY$, $(0, -4, 5)$ on $YZ$, $(3, 0, 5)$ on $ZX$.

4
Worked Example
The point $(x, y, z)$ lies in octant VI. Write the sign of each coordinate, and give one such point.
Solution
  1. From the octant table, octant VI has the sign pattern $(-, +, -)$.
  2. So $x < 0$, $y > 0$ and $z < 0$.
  3. A convenient example is $(-1, 4, -2)$.

Answer: signs $(-, +, -)$; one such point is $(-1, 4, -2)$.

5
Worked Example
A point $P$ has $x$-coordinate $6$, $z$-coordinate $-2$ and lies on the $ZX$-plane. Find $P$.
Solution
  1. The $ZX$-plane is defined by $y = 0$.
  2. So the $y$-coordinate of $P$ must be $0$.
  3. Combine with the given $x = 6$ and $z = -2$.

Answer: $P = (6, 0, -2)$.

6
Worked Example
Find the image (reflection) of the point $A(2, 3, 4)$ in (a) the $XY$-plane and (b) the $x$-axis.
Solution
  1. (a) Reflecting in the $XY$-plane reverses the sign of $z$ only: $(2, 3, -4)$.
  2. (b) Reflecting in the $x$-axis keeps $x$ but reverses both $y$ and $z$: $(2, -3, -4)$.

Answer: (a) $(2, 3, -4)$; (b) $(2, -3, -4)$.

Key Points

  • A point in space needs an ordered triple $(x, y, z)$; the three axes meet at the origin $O(0,0,0)$.
  • The three coordinate planes are $XY$ ($z=0$), $YZ$ ($x=0$) and $ZX$ ($y=0$).
  • One zero coordinate places a point on a coordinate plane; two zeros place it on an axis.
  • The planes cut space into 8 octants; octant I is $(+,+,+)$ and octants I–IV sit above the $XY$-plane, V–VIII below it.
Quick Check — 4 MCQs
Tap an option to check your answer0 / 4
Q1.The number of octants in space is:
Explanation: Three coordinate planes divide space into $8$ octants.
Q2.The coordinates of the origin are:
Explanation: All three coordinates are zero.
Q3.On the $xy$-plane, the $z$-coordinate is:
Explanation: Points on the $xy$-plane have $z=0$.
Q4.The point $(1,2,3)$ lies in the:
Explanation: All coordinates positive $\Rightarrow$ first octant.
Practice more MCQs → 📝 Assignment · 35 marks