Chapter 11 · Introduction to Three Dimensional Geometry

11.1 Coordinate Axes, Coordinate Planes and Octants

Three mutually perpendicular axes through one point $O$ split space by three coordinate planes into eight octants, and every point $P$ gets a unique address $(x,y,z)$ — its signed distances from the three planes.

A seat in a cinema hall is found with two labels — a row letter and a seat number — because the seats all lie on one floor. A cricket ball in flight is different. To say where it is at one instant, the Hawk-Eye system that umpires use for decisions needs three numbers: how far down the pitch, how far across it, and how high above the ground. A flat sheet of paper has two directions to move in; the room you are sitting in has three. Any honest description of a point in space must therefore use three numbers, not two.

In Class 10 you located points in a plane with two perpendicular number lines, the $x$-axis and the $y$-axis. In this section we add a third number line, perpendicular to both. The three axes pair up into three flat coordinate planes, the planes cut space into eight octants (just as two axes cut the plane into four quadrants), and every point receives an address $(x,y,z)$. The whole section is about reading and writing those addresses correctly — which signs, which zeros, and which point is the foot of which perpendicular.

Three Mutually Perpendicular Axes

Pick a point $O$ in space — a corner of the room will do. Through $O$ draw three straight lines, each perpendicular to the other two: think of the line where the floor meets the left wall, the line where the floor meets the back wall, and the vertical line where the two walls meet. Mark each line as a number line with $0$ at $O$ and the same unit of length on all three. These are the coordinate axes.

📘 Definition 11.1 — Rectangular coordinate system in space

A rectangular coordinate system in space consists of a point $O$, the origin, and three mutually perpendicular number lines through $O$ with a common unit: the $x$-axis $X'OX$, the $y$-axis $Y'OY$ and the $z$-axis $Z'OZ$. The rays $OX$, $OY$, $OZ$ are the positive directions and $OX'$, $OY'$, $OZ'$ the negative ones.

The system is right-handed: if the fingers of your right hand curl from $OX$ towards $OY$ (through $90^\circ$), your thumb points along $OZ$.

The right-hand rule fixes which way the $z$-axis points once the $x$- and $y$-axes are drawn. In the usual picture the $x$-axis comes out of the page towards you (drawn slanting down to the left), the $y$-axis runs to the right and the $z$-axis points up. If you draw $OZ$ downwards instead, you get a left-handed system — a mirror image — and every sign rule in this chapter would flip. Always draw your axes right-handed.

📘 Definition 11.2 — Coordinate planes

Each pair of axes fixes a plane through $O$; these three planes are the coordinate planes:

the $XY$-plane, containing the $x$- and $y$-axes (the "floor");
the $YZ$-plane, containing the $y$- and $z$-axes;
the $ZX$-plane, containing the $z$- and $x$-axes.

The three coordinate planes are mutually perpendicular, and any two of them meet in an axis: the $XY$- and $ZX$-planes meet in the $x$-axis, the $XY$- and $YZ$-planes in the $y$-axis, and the $YZ$- and $ZX$-planes in the $z$-axis.

A useful way to remember which axis is perpendicular to which plane: the plane is named by two letters, and the missing letter is the axis that sticks straight out of it. The $z$-axis is perpendicular to the $XY$-plane, the $x$-axis to the $YZ$-plane and the $y$-axis to the $ZX$-plane.

Coordinates of a Point in Space

Now take any point $P$. Drop a perpendicular $PM$ from $P$ to the $XY$-plane, meeting it at $M$, and from $M$ drop a perpendicular $MA$ to the $x$-axis. The three segments $OA$, $AM$ and $MP$ run parallel to the $x$-, $y$- and $z$-axes. Their signed lengths are the coordinates of $P$. The same three numbers can be read off another way: they are the signed distances of $P$ from the three coordinate planes.

📘 Definition 11.3 — Coordinates of a point in space

The coordinates of a point $P$ are the three real numbers

$x$ = the distance of $P$ from the $YZ$-plane, taken positive on the side of $OX$ and negative on the side of $OX'$;
$y$ = the distance of $P$ from the $ZX$-plane, positive on the side of $OY$, negative on the side of $OY'$;
$z$ = the distance of $P$ from the $XY$-plane, positive on the side of $OZ$, negative on the side of $OZ'$.

We write $P(x,y,z)$, always in this order. Equivalently, the planes through $P$ parallel to the $YZ$-, $ZX$- and $XY$-planes cut the $x$-, $y$- and $z$-axes at the points $x$, $y$ and $z$ of their number lines.

Going backwards is just as easy. To plot $P(3,-2,4)$: walk $3$ units from $O$ along $OX$ to $A(3,0,0)$, then $2$ units parallel to $OY'$ (because $y$ is negative) to $M(3,-2,0)$ on the floor, then $4$ units straight up to $P$. You have walked along three edges of a rectangular box that has $O$ and $P$ as opposite corners. That box — with edges $|x|$, $|y|$ and $|z|$ — is the picture to keep in your head for every point in this chapter.

📐 Theorem 11.1 — Points and ordered triples

Once a rectangular coordinate system is fixed, each point of space has exactly one ordered triple $(x,y,z)$ of real numbers as its coordinates, and each ordered triple is the coordinates of exactly one point. In particular:

Where the point lies	Its coordinates	Condition
origin $O$	$(0,0,0)$	$x=y=z=0$
$x$-axis	$(x,0,0)$	$y=0,\ z=0$
$y$-axis	$(0,y,0)$	$z=0,\ x=0$
$z$-axis	$(0,0,z)$	$x=0,\ y=0$
$XY$-plane	$(x,y,0)$	$z=0$
$YZ$-plane	$(0,y,z)$	$x=0$
$ZX$-plane	$(x,0,z)$	$y=0$

So $z=0$ is the equation of the $XY$-plane, $x=0$ that of the $YZ$-plane and $y=0$ that of the $ZX$-plane.

Why. A point lies on the $XY$-plane exactly when its distance from that plane is $0$, that is when $z=0$; the other two planes are the same. A point lies on the $x$-axis exactly when it lies on both the $XY$-plane and the $ZX$-plane (they meet in the $x$-axis), that is when $z=0$ and $y=0$. The one-to-one correspondence holds because the three perpendicular distances can be measured from any point, and the "walk along three edges" construction reaches exactly one point from any triple. $\blacksquare$

In the language of Section 2.1, space is now a copy of $\mathbb{R}\times\mathbb{R}\times\mathbb{R}$, the set of all ordered triples of real numbers. For the same reason $(1,2,3)$ and $(3,2,1)$ are different points: the order of the coordinates matters.

Feet of perpendiculars. Dropping a perpendicular from $P(x,y,z)$ to a coordinate plane changes only the coordinate measured from that plane, which becomes $0$. Dropping a perpendicular to an axis keeps only that axis's coordinate:

Perpendicular from $P(x,y,z)$ to	$XY$-plane	$YZ$-plane	$ZX$-plane	$x$-axis	$y$-axis	$z$-axis
foot of the perpendicular	$(x,y,0)$	$(0,y,z)$	$(x,0,z)$	$(x,0,0)$	$(0,y,0)$	$(0,0,z)$

These six feet, together with $O$ and $P$, are exactly the eight corners of the box with opposite corners $O$ and $P$. The distance of $P$ from the $XY$-plane is $|z|$ (a distance is never negative), from the $YZ$-plane $|x|$, and from the $ZX$-plane $|y|$.

✏️ Example 1 — Coordinates from distances

A point $P$ lies $4$ units from the $YZ$-plane on the side of $OX$, $2$ units from the $ZX$-plane on the side of $OY'$, and $5$ units below the $XY$-plane.
(a) Write the coordinates of $P$.
(b) Find the feet of the perpendiculars from $P$ to the three coordinate planes and to the three axes.

💡 Show solution▶

(a) The distance from the $YZ$-plane gives $x$: on the $OX$ side, so $x=+4$. The distance from the $ZX$-plane gives $y$: on the $OY'$ side, so $y=-2$. Below the $XY$-plane means on the $OZ'$ side, so $z=-5$. Hence $P(4,-2,-5)$.

(b) planes Put a $0$ in the place of the coordinate measured from each plane: $XY$-plane $(4,-2,0)$; $YZ$-plane $(0,-2,-5)$; $ZX$-plane $(4,0,-5)$.

(b) axes Keep only one coordinate: $x$-axis $(4,0,0)$; $y$-axis $(0,-2,0)$; $z$-axis $(0,0,-5)$.

Check Each foot on a plane satisfies that plane's equation ($z=0$, $x=0$, $y=0$ respectively), and each foot differs from $P$ in exactly one coordinate, so the segment joining it to $P$ is parallel to one axis — perpendicular to the plane ✓.

✏️ Example 2 — The corners of a box

A rectangular box (cuboid) has one corner at the origin $O$ and three edges $OA=5$, $OB=3$, $OC=2$ along the positive $x$-, $y$- and $z$-axes. Write the coordinates of all eight corners. Which corners lie in the $XY$-plane, which in the $YZ$-plane, and which corner is not on any coordinate plane?

💡 Show solution▶

Axes $O(0,0,0)$, $A(5,0,0)$, $B(0,3,0)$, $C(0,0,2)$.

Faces The other corners are reached by moving along two or three edge directions: $D(5,3,0)$, $E(0,3,2)$, $F(5,0,2)$ and the far corner $P(5,3,2)$.

Planes $XY$-plane ($z=0$): $O,A,B,D$ — the bottom face. $YZ$-plane ($x=0$): $O,B,C,E$ — the back face. (The $ZX$-plane, $y=0$, holds $O,A,C,F$.)

Answer Only $P(5,3,2)$, the corner opposite $O$, has all three coordinates non-zero, so it lies on no coordinate plane.

✏️ Example 3 — What does an equation describe?

Describe in words the set of all points $(x,y,z)$ in space for which
(a) $z=4$; (b) $y=-2$; (c) $x=0$ and $z=0$; (d) $x=3$ and $y=2$.

💡 Show solution▶

(a) Every such point is $4$ units above the $XY$-plane, and $x$, $y$ are free. This is the plane parallel to the $XY$-plane, $4$ units above it — like a ceiling $4$ units over the floor.

(b) The plane parallel to the $ZX$-plane, $2$ units from it on the side of $OY'$.

(c) $x=0$ is the $YZ$-plane and $z=0$ the $XY$-plane; both together give their common line, the $y$-axis — the points $(0,y,0)$.

(d) Only $z$ is free: the points $(3,2,z)$ form the vertical line through $(3,2,0)$, parallel to the $z$-axis.

Note In the plane, $x=3$ was a line; in space the same equation is a whole plane. One equation in space usually describes a surface, and two equations together a line.

The point $P(x,y,z)$ is the far corner of a box whose opposite corner is $O$. Coloured dashed lines drop perpendiculars from $P$ to the $XY$-plane (green), the $YZ$-plane (blue) and the $ZX$-plane (red); their lengths are $|z|$, $|x|$ and $|y|$. Use the sliders, or drag $P$ itself (choose what dragging changes); drag anywhere else to turn the axes. Play sends $P$ on a tour of all eight octants.

$x$3$y$2$z$2.5drag $P$ changesview

$P$—

foot on $XY$-plane—

foot on $YZ$-plane—

foot on $ZX$-plane—

octant—

The Eight Octants

Two axes cut the plane into four quadrants. Three coordinate planes cut space into eight pieces: the $XY$-plane separates "above" ($z\gt0$) from "below" ($z\lt0$), the $YZ$-plane separates $x\gt0$ from $x\lt0$, and the $ZX$-plane separates $y\gt0$ from $y\lt0$. Each choice of three signs gives one piece, and $2\times2\times2=8$.

📘 Definition 11.4 — Octants

The three coordinate planes divide space into eight regions called octants. Each octant is named by the three rays that bound it, and numbered I to VIII:

Octant	I	II	III	IV	V	VI	VII	VIII
Name	$XOYZ$	$X'OYZ$	$X'OY'Z$	$XOY'Z$	$XOYZ'$	$X'OYZ'$	$X'OY'Z'$	$XOY'Z'$
$x$	$+$	$-$	$-$	$+$	$+$	$-$	$-$	$+$
$y$	$+$	$+$	$-$	$-$	$+$	$+$	$-$	$-$
$z$	$+$	$+$	$+$	$+$	$-$	$-$	$-$	$-$

A point whose coordinates are all non-zero lies in exactly one octant, read off from its signs. A point with at least one coordinate equal to $0$ lies on a coordinate plane and belongs to no octant.

How to remember the numbering. Octants I–IV lie above the $XY$-plane ($z\gt0$). Looking down on the floor, their $(x,y)$ signs go round exactly like the quadrants of Class 10: $(+,+)$, $(-,+)$, $(-,-)$, $(+,-)$. Octants V–VIII lie directly below I–IV, in the same order, with $z\lt0$: octant V is under octant I, VI under II, VII under III and VIII under IV. In each octant's name, a prime ($X'$, $Y'$, $Z'$) marks a negative coordinate.

✏️ Example 4 — Naming octants

In which octant, if any, does each point lie?
(a) $(2,-3,5)$ (b) $(-1,-4,-6)$ (c) $(-3,2,-1)$ (d) $(4,1,-2)$ (e) $(-5,3,7)$ (f) $(0,3,-2)$

💡 Show solution▶

(a) Signs $(+,-,+)$: above the floor, and $(+,-)$ is the fourth quadrant — octant IV, $XOY'Z$.

(b) $(-,-,-)$: octant VII, $X'OY'Z'$.

(d) $(+,+,-)$: octant V, $XOYZ'$ (directly below octant I).

(e) $(-,+,+)$: octant II, $X'OYZ$.

(f) $x=0$, so the point lies on the $YZ$-plane — in no octant. It sits on the "wall" separating octant V, signs $(+,+,-)$, from octant VI, signs $(-,+,-)$.

✏️ Example 5 — Mirror images in the coordinate planes

A plane mirror is placed along each coordinate plane in turn. Find the image of $P(3,-2,4)$ in (a) the $XY$-plane, (b) the $YZ$-plane, (c) the $ZX$-plane, and (d) the point $P'$ on the line $PO$ with $O$ halfway between $P$ and $P'$ (the "reflection in the origin"). Name the octant of $P$ and of each image.

💡 Show solution▶

Idea The image in a plane mirror is as far behind the mirror as the object is in front, on the same perpendicular. Only the coordinate measured from the mirror plane changes — it changes sign. $P$ itself has signs $(+,-,+)$: octant IV.

(a), (b), (c) $XY$-plane: $(3,-2,-4)$, signs $(+,-,-)$, octant VIII. $\ YZ$-plane: $(-3,-2,4)$, signs $(-,-,+)$, octant III. $\ ZX$-plane: $(3,2,4)$, signs $(+,+,+)$, octant I.

(d) Going through $O$ to the same distance on the other side reverses every direction: $P'(-3,2,-4)$, signs $(-,+,-)$, octant VI.

Check In each of (a)–(c) the point and its image have the same foot on the mirror plane (for (a) both feet are $(3,-2,0)$) and equal distances from it ✓. Reflecting in all three planes one after another also gives $(-3,2,-4)$ ✓.

✏️ Example 6 — A drone over a stadium

At a stadium, the ground is taken as the $XY$-plane with the origin at the foot of a floodlight tower, the $x$-axis pointing east, the $y$-axis pointing north and the $z$-axis vertically up; the unit is $1\ \text{m}$. A camera drone hovers $120\ \text{m}$ east of the tower, $80\ \text{m}$ south of it, at a height of $45\ \text{m}$.
(a) Write its coordinates and name its octant.
(b) At noon the Sun is directly overhead. Where is the drone's shadow?
(c) A second drone hovers at the mirror image of the first in the vertical plane through the tower that runs east–west. Find its coordinates and its height above the ground.

💡 Show solution▶

(a) East is $+x$, south is $-y$, up is $+z$: $D(120,-80,45)$. Signs $(+,-,+)$: octant IV.

(b) Vertical sunlight casts the shadow at the foot of the perpendicular to the ground: $(120,-80,0)$, $120\ \text{m}$ east and $80\ \text{m}$ south of the tower.

(c) The vertical east–west plane through the tower contains the $x$- and $z$-axes: it is the $ZX$-plane, $y=0$. The image changes the sign of $y$: $(120,80,45)$ — $80\ \text{m}$ north of the tower and still $45\ \text{m}$ high, in octant I.

The three coordinate planes are drawn as see-through squares, and one octant is filled in as a cube with $O$ as a corner. Press Play to visit octants I to VIII in turn and watch the sign pattern change, or pick one yourself. Then press 🎲 for a mystery point $Q$ and change the octant until the verdict says $Q$ is inside. Drag to turn the picture.

octantview

octant—

signs of $(x,y,z)$—

a sample point—

mystery point $Q$—

⚠️ Common mistakes with coordinates in space

1. Mixing up "distance from a plane" and "coordinate along an axis". The distance of $P(x,y,z)$ from the $XY$-plane is $|z|$, not $|x|$ or $|y|$: the plane named by two letters is measured from by the third coordinate.

2. Forgetting the absolute value. $P(2,-6,-3)$ is $3$ units from the $XY$-plane, not $-3$. Coordinates can be negative; distances cannot.

3. Putting the zero in the wrong place. The foot of the perpendicular from $(5,-3,7)$ to the $ZX$-plane is $(5,0,7)$ — the $y$-coordinate becomes $0$, because the $ZX$-plane is $y=0$.

4. Giving a point on a plane an octant. $(0,3,-2)$ is not in octant V or VI; it is on the $YZ$-plane, the wall between them.

5. Thinking $x=3$ is a line. In space, $x=3$ is a whole plane parallel to the $YZ$-plane. You need two equations, such as $x=3,\ y=2$, to describe a line.

Exercises

Coordinates and feet of perpendiculars.

1 Write the coordinates of the point that is $6$ units from the $YZ$-plane on the side of $OX'$, $1$ unit from the $ZX$-plane on the side of $OY$, and $3$ units above the $XY$-plane.
2 Find the distances of $P(-2,5,-7)$ from the $XY$-, $YZ$- and $ZX$-planes.
3 Find the feet of the perpendiculars from $Q(6,-4,9)$ to the three coordinate planes.
4 Find the feet of the perpendiculars from $Q(6,-4,9)$ to the three coordinate axes.
5 What are the $y$- and $z$-coordinates of every point on the $x$-axis? Write the general form of a point on the $z$-axis.
6 Of the points $A(0,4,0)$, $B(3,0,-1)$, $C(0,0,-5)$, $D(2,-1,0)$, $E(0,-3,6)$, which lie (i) on a coordinate axis, (ii) in the $XY$-plane, (iii) in the $YZ$-plane, (iv) in the $ZX$-plane?
7 The point $(a-3,\ 2b+4,\ c)$ lies on the $z$-axis. Find $a$ and $b$. What can you say about $c$?
8 The foot of the perpendicular from a point $P$ to the $XY$-plane is $(2,-5,0)$, and $P$ is $4$ units below the $XY$-plane. Find $P$ and the foot of the perpendicular from $P$ to the $YZ$-plane.

Check: octant, feet and distances of any point

$x$$y$$z$

octant—

feet on $XY$, $YZ$, $ZX$—

distances from $XY$, $YZ$, $ZX$—

💡 Show solutions▶

1 $x=-6$ (side of $OX'$), $y=1$, $z=3$: the point $(-6,1,3)$.

2 From the $XY$-plane $|z|=7$; from the $YZ$-plane $|x|=2$; from the $ZX$-plane $|y|=5$.

3 $XY$-plane $(6,-4,0)$; $YZ$-plane $(0,-4,9)$; $ZX$-plane $(6,0,9)$.

4 $x$-axis $(6,0,0)$; $y$-axis $(0,-4,0)$; $z$-axis $(0,0,9)$.

5 Both are $0$: a point on the $x$-axis is $(a,0,0)$. A point on the $z$-axis is $(0,0,c)$ for some real $c$.

6 (i) $A$ ($y$-axis) and $C$ ($z$-axis). (ii) $z=0$: $A$, $D$. (iii) $x=0$: $A$, $C$, $E$. (iv) $y=0$: $B$, $C$. (A point on an axis lies on two coordinate planes.)

7 On the $z$-axis $x=0$ and $y=0$: $a-3=0$, $a=3$; $2b+4=0$, $b=-2$. $c$ can be any real number.

8 $P$ has the same $x$ and $y$ as its foot, and $z=-4$: $P(2,-5,-4)$. Its foot on the $YZ$-plane is $(0,-5,-4)$.

Planes and lines described by equations.

1 Write the equation of the plane parallel to the $XY$-plane and $3$ units below it.
2 Write the equation of the plane through $(5,-2,8)$ parallel to the $YZ$-plane.
3 Describe the set $\{(x,y,z) : y=0,\ z=0\}$.
4 Describe the set of points with $x=-1$ and $z=2$.
5 Which coordinate plane is perpendicular to the $y$-axis? Write its equation.
6 The planes $x=2$ and $y=-3$ meet in a line. Write the general point of this line and the point where it crosses the $XY$-plane.
7 How many points satisfy all three equations $x=1$, $y=2$, $z=3$? How many satisfy just the first two?

💡 Show solutions▶

1 $z=-3$.

2 Every point of the plane has the same $x$ as $(5,-2,8)$: $x=5$.

3 The points $(x,0,0)$: the $x$-axis.

4 The points $(-1,y,2)$, $y$ free: the line through $(-1,0,2)$ parallel to the $y$-axis.

5 The $ZX$-plane (the plane without the letter $Y$); its equation is $y=0$.

6 General point $(2,-3,t)$, $t$ any real number — a line parallel to the $z$-axis. It crosses the $XY$-plane ($z=0$) at $(2,-3,0)$.

7 Exactly one point, $(1,2,3)$. Infinitely many — the line $(1,2,t)$ parallel to the $z$-axis.

III

Octants and mirror images.

1 Name the octant of $(1,-2,-3)$.
2 Name the octant of $(-4,-2,5)$.
3 Name the octant of $(-3,1,-2)$.
4 Name the octant of $(5,7,2)$, and of $(5,7,0)$.
5 Write the sign pattern of the coordinates in octant $X'OYZ'$, and its number.
6 Find the images of $(-2,5,3)$ in the $XY$-, $YZ$- and $ZX$-planes.
7 Find the reflection of $(4,-1,-6)$ in the origin, and name the octants of the point and of its reflection.
8 (a) Find all real $k$ for which $(k,\ k-4,\ 2-k)$ lies in octant IV. (b) Show that this point is never in octant I.

💡 Show solutions▶

1 $(+,-,-)$: octant VIII.

2 $(-,-,+)$: octant III.

3 $(-,+,-)$: octant VI.

4 $(5,7,2)$: octant I. $(5,7,0)$ has $z=0$: it lies in the $XY$-plane, in no octant.

5 Primes mark negatives: $(-,+,-)$, octant VI.

6 $XY$-plane $(-2,5,-3)$; $YZ$-plane $(2,5,3)$; $ZX$-plane $(-2,-5,3)$.

7 $(-4,1,6)$. The point has signs $(+,-,-)$: octant VIII; its reflection $(-,+,+)$: octant II.

8 (a) Octant IV needs $x\gt0$, $y\lt0$, $z\gt0$: $k\gt0$, $k\lt4$, $k\lt2$, so $0\lt k\lt2$. (b) Octant I needs $k\gt0$, $k\gt4$ and $k\lt2$; $k\gt4$ and $k\lt2$ cannot both hold.

Boxes and applications.

1 A cuboid has one corner at $O$ and edges of lengths $4$, $6$ and $3$ along the negative $x$-axis, the positive $y$-axis and the negative $z$-axis. Write its eight corners. Which octant contains its interior?
2 A cube of edge $2$ has its centre at $O$ and its edges parallel to the axes. Write its eight corners. How many octants contain a corner?
3 A box with edges parallel to the axes has opposite corners $(1,2,3)$ and $(4,6,8)$. Write its other six corners and the lengths of its edges.
4 A classroom is $6\ \text{m}$ long, $4\ \text{m}$ wide and $3\ \text{m}$ high. The origin is a corner of the floor, the $x$-axis runs along a $6\ \text{m}$ wall, the $y$-axis along a $4\ \text{m}$ wall and the $z$-axis up. (a) A fan hangs from the centre of the ceiling with its blades $0.5\ \text{m}$ below the ceiling; give the coordinates of the centre of the blades. (b) A light switch is on the wall in the $YZ$-plane, $0.8\ \text{m}$ from the corner line $x=y=0$ and $1.2\ \text{m}$ above the floor. Give its coordinates.
5 An airport radar uses $x$ east, $y$ north and $z$ up, in kilometres. An aircraft is $12\ \text{km}$ west and $5\ \text{km}$ north of the radar at an altitude of $3\ \text{km}$. Give its coordinates and octant, and the point on the ground directly below it.
6 A point in octant VII is at distance $3$ from each of the three coordinate planes. Find it.
7 Each coordinate of a point is chosen from $\{-1,0,1\}$. How many such points are there, and how many of them lie in an octant?

💡 Show solutions▶

1 $O(0,0,0)$, $(-4,0,0)$, $(0,6,0)$, $(0,0,-3)$, $(-4,6,0)$, $(0,6,-3)$, $(-4,0,-3)$, $(-4,6,-3)$. Inside, $x\lt0$, $y\gt0$, $z\lt0$: octant VI.

2 The corners are the eight points $(\pm1,\pm1,\pm1)$ — every choice of signs once — so each of the $8$ octants contains exactly one corner.

3 Each coordinate is one of the two values of the opposite corners: $(4,2,3)$, $(1,6,3)$, $(1,2,8)$, $(4,6,3)$, $(4,2,8)$, $(1,6,8)$. Edges $4-1=3$, $6-2=4$, $8-3=5$.

4 (a) Centre of the ceiling is above $(3,2,0)$; the blades are at height $3-0.5=2.5$: $(3,\ 2,\ 2.5)$. (b) On the $YZ$-plane $x=0$: $(0,\ 0.8,\ 1.2)$.

5 West is $-x$: $(-12,5,3)$, signs $(-,+,+)$, octant II. Directly below: $(-12,5,0)$.

6 $|x|=|y|=|z|=3$ and all negative (octant VII): $(-3,-3,-3)$.

7 $3\times3\times3=27$ points. In an octant every coordinate is non-zero, i.e. $\pm1$: $2\times2\times2=8$ points (one per octant). The other $19$ lie on coordinate planes.

🎯 Exam corner Board · 1 markBoard · 2 marksBoard · Case studyJEE Main

How this is examined: Board papers ask 1-mark questions on octants, the equation of a coordinate plane or the foot of a perpendicular, and 2-mark questions combining feet and distances; case studies place objects in a room or a warehouse. In JEE these ideas are the language of the later three-dimensional geometry problems, and appear on their own as sign-pattern and counting questions.

E1. Board · 1 mark The point $(-2,3,-4)$ lies in octant (A) II (B) VI (C) VII (D) III

💡 Show solution▶

Idea Signs $(-,+,-)$. Below the $XY$-plane ($z\lt0$) means octants V–VIII; $(x,y)$ signs $(-,+)$ are the second quadrant, so the octant is the one under II.

Trap (A) II has the right $x$ and $y$ signs but ignores $z\lt0$.

Answer (B) VI, $X'OYZ'$.

E2. Board · 1 mark Assertion–Reason. Assertion (A): the point $(4,0,-2)$ lies in the $ZX$-plane. Reason (R): every point of the $ZX$-plane has $y$-coordinate $0$. (A) Both A and R are true and R is the correct explanation of A. (B) Both are true but R is not the correct explanation of A. (C) A is true but R is false. (D) A is false but R is true.

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Idea The $ZX$-plane is $y=0$ (R is true), and $(4,0,-2)$ has $y=0$, so it lies in that plane (A is true) — precisely because of R.

Answer (A).

E3. Board · 2 marks Find the feet of the perpendiculars from $P(-3,7,-2)$ to the three coordinate planes, and the distance of $P$ from each plane.

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Feet $XY$-plane $(-3,7,0)$, $YZ$-plane $(0,7,-2)$, $ZX$-plane $(-3,0,-2)$. (1 mark)

Distances From the $XY$-plane $|z|=2$, from the $YZ$-plane $|x|=3$, from the $ZX$-plane $|y|=7$. (1 mark)

E4. Board · Case study In an automated warehouse the origin is a corner of the floor, the $x$- and $y$-axes run along the two walls that meet there and the $z$-axis points up; the unit is $1\ \text{m}$, and the building occupies only points with $x\ge0$, $y\ge0$, $z\ge0$. A carton sits at $A(8,5,2)$. (i) Give the point of the floor directly below $A$. (1 mark) (ii) A robot arm lifts the carton $1.5\ \text{m}$ straight up and then moves it $3\ \text{m}$ in the direction of the negative $x$-axis. Where is it now? (1 mark) (iii) A technician's log lists the points $(0,4,3)$, $(6,-1,2)$, $(3,3,0)$ and $(-2,4,3)$. Which can be inside the building or on its walls or floor? For each of those on a coordinate plane, name the plane. (2 marks)

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(i) Same $x$ and $y$, $z=0$: $(8,5,0)$. (1 mark)

(ii) Up $1.5$: $(8,5,3.5)$; then $3$ units towards $OX'$: $(5,5,3.5)$. (1 mark)

(iii) Test signs $(6,-1,2)$ has $y\lt0$ (octant IV) and $(-2,4,3)$ has $x\lt0$ (octant II): both are outside. (1 mark)

(iii) Answer $(0,4,3)$ is on the wall $x=0$, the $YZ$-plane; $(3,3,0)$ is on the floor $z=0$, the $XY$-plane. (1 mark)

E5. JEE Main The set of all real $k$ for which the point $(k-2,\ 3-k,\ k-5)$ lies in octant VIII is (A) $(2,3)$ (B) $(3,5)$ (C) $(2,5)$ (D) $(5,\infty)$

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Idea Octant VIII is $XOY'Z'$: $x\gt0$, $y\lt0$, $z\lt0$. So $k\gt2$, $3-k\lt0$ i.e. $k\gt3$, and $k\lt5$.

Combine All three: $3\lt k\lt5$.

Trap (C) uses only the conditions on $x$ and $z$; at $k=2.5$ the point is $(0.5,0.5,-2.5)$, in octant V.

Answer (B).

E6. JEE Main Numerical answer. Consider all points with integer coordinates satisfying $|x|+|y|+|z|=3$. How many of them lie on at least one coordinate plane?

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Count all Split by the number of non-zero coordinates. One non-zero coordinate: it is $\pm3$ in one of $3$ places — $6$ points. Two non-zero: choose the zero place ($3$ ways), the absolute values $(1,2)$ or $(2,1)$ ($2$ ways) and the signs ($4$ ways) — $24$ points. Three non-zero: all equal to $\pm1$ — $8$ points. Total $38$.

In octants A point lies in an octant exactly when no coordinate is $0$: the $8$ points $(\pm1,\pm1,\pm1)$.

Answer $38-8=30$.

⏱ Fast route The points with a zero coordinate are exactly the $6+24$ counted in the first two cases: $30$.

🔬 Research–Teaching NexusMini project · about 45 minutes

Research question. Can you build a three-dimensional "address book" for your classroom, so that anyone given only the coordinates could walk up and touch each object?

Why it matters. Warehouses, hospitals and factories store the positions of shelves, sensors and robot paths as $(x,y,z)$ triples; architects' building models and 3-D printers work the same way. A coordinate system is only useful if everyone agrees on the origin, the directions of the axes, the unit and the right-hand rule.

What to do.

Choose a floor corner of the room as $O$. Decide which wall line is the $x$-axis and which the $y$-axis so that the $z$-axis (up) makes a right-handed system. Draw a labelled sketch.
With a measuring tape, find the coordinates (in metres, to the nearest centimetre) of at least eight objects: a light, the fan, a window catch, the top of the board, a door handle, your bag hook, a switch, a projector.
Write each object's feet on the three coordinate planes and say which objects lie on a coordinate plane (on a wall or the floor).
Swap your list of coordinates — without names — with a classmate, and let them find each object. Record which ones were found correctly and why the others were not.
Now move the origin to the opposite floor corner. How do all the coordinates change? Find a rule.

Tools: A measuring tape (or a phone's measuring app), paper for sketches, a spreadsheet

Hand in: Your sketch with axes, the table of coordinates and feet, the swap results, and your rule for changing the origin.

Going further. Find out how a phone stores its position in three numbers (latitude, longitude and height above sea level). Why is that not a rectangular coordinate system, and over what size of region does it behave almost like one?

Instructor note: the most common error is a left-handed set of axes — check every sketch with the right-hand rule. Moving the origin to the corner diagonally opposite on the floor (room $L\times W$) sends $(x,y,z)$ to $(L-x,\ W-y,\ z)$ if both horizontal axes are reversed; a pair that reverses only one axis becomes left-handed, which makes a good discussion. Latitude/longitude/height is a curved (spherical-type) system; it is close to rectangular over a few hundred metres.

🧪 Self-check quiz

0/5 correct

Q1. The point $(-3,4,-2)$ lies in octant

Q2. The equation of the $YZ$-plane is

Q3. The foot of the perpendicular from $(5,-3,7)$ to the $ZX$-plane is

Q4. Every point on the $y$-axis has the form

Q5. The distance of $P(2,-6,-3)$ from the $XY$-plane is

← Chapter 11 — Introduction to Three Dimensional Geometry 11.2 Distance between Two Points →