Chapter 11 · Introduction

Chapter 11 — Introduction to Three Dimensional Geometry

A point on a line needs one number and a point on a page needs two; a point in a room, in the sky or in orbit needs three. Add a third axis, meet the eight octants, and measure straight-line distances in space.

A well-known story says that the French thinker René Descartes, lying in bed one morning, watched a fly crawling on the ceiling and wondered how to describe exactly where it was. His answer: give its distances from two walls that meet in a corner. Two numbers fix a point on a flat surface — that idea grew into the coordinate geometry of Class 10 and Chapters 9 and 10 of this course. But when the fly takes off and hovers in the middle of the room, two numbers are no longer enough. You also need its height above the floor. Three numbers — the distance from one wall, the distance from the other wall and the height — pin the fly down exactly.

The same three numbers are everywhere around you. An air-traffic controller at Delhi airport tracks every aircraft by two map coordinates and an altitude. Your phone's GPS reports latitude, longitude and height. A 3-D printer builds a model by moving its nozzle to one point $(x,y,z)$ after another, and a robot arm in a car factory is programmed the same way. In this short chapter you will build the coordinate system that makes all this possible: three mutually perpendicular coordinate axes, the three coordinate planes they form in pairs, and the eight octants into which those planes cut space. Then you will extend the Pythagoras-based distance formula from the plane to space, and use it to test whether points are collinear, to classify triangles and to describe surfaces such as spheres by equations.

Three perpendicular axes through $O$→Three coordinate planes, eight octants→A point is an ordered triple $P(x,y,z)$→Distance $PQ=\sqrt{\Delta x^2+\Delta y^2+\Delta z^2}$

🎯 By the end of this chapter you can…

draw a right-handed system of coordinate axes in space and name the $XY$-, $YZ$- and $ZX$-planes together with their equations $z=0$, $x=0$ and $y=0$;
find the coordinates $(x,y,z)$ of a point as its signed perpendicular distances from the three coordinate planes, and plot a point from its coordinates;
write down the coordinates of points on the axes and on the coordinate planes, and the feet of the perpendiculars from a point to the axes and planes;
name the octant containing a given point from the signs of its coordinates, and the sign pattern of any octant;
find the distance between two points in space, and use it to show that points are collinear or form an isosceles, equilateral or right-angled triangle, and to write the equation of a set of points satisfying a distance condition.

The room is $5\ \text{m}$ long, $4\ \text{m}$ wide and $3\ \text{m}$ high, and the origin $O$ is the corner where the floor meets the two back walls. A small lamp (or a hovering drone) $L$ floats in the room. The dashed lines drop perpendiculars from $L$ to the two walls and to the floor — their lengths are exactly the three coordinates of $L$. Press Play to let it drift, move the sliders yourself, and drag the picture (or use the view slider) to walk round the room.

$x$3$y$2$z$1.5view

from wall $x=0$—

from wall $y=0$—

height above floor—

lamp $L$—

straight line $OL$—

Sections in this chapter

11.1

Coordinate Axes, Coordinate Planes and Octants

Three mutually perpendicular axes, the right-hand rule, the coordinate planes and their equations, the coordinates of a point in space, and the sign patterns of the eight octants.

$P(x,y,z)$, $\ XY$-plane: $z=0$

Open section →11.2

Distance between Two Points

The distance formula in space as the diagonal of a box, distances from the origin and from the axes, collinearity, special triangles and quadrilaterals, and sets of points defined by distances.

$PQ=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2}$

Open section →

✅ Before you start

From Class 10 Mathematics you need coordinate geometry in the plane (Chapter 7 — plotting points, the four quadrants and the distance formula $\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$) and Pythagoras' theorem, which this chapter uses twice in a row. Chapter 9 of this course (Straight Lines) recalls the plane distance formula as well. From this course you need ordered pairs and ordered triples from Section 2.1 — a point in space will be an ordered triple, an element of $\mathbb{R}\times\mathbb{R}\times\mathbb{R}$ — and the language of sets from Section 1.1, since a plane or a sphere will be described as a set of points $\{(x,y,z) : \dots\}$. Simplifying surds such as $\sqrt{50}=5\sqrt2$ and expanding $(a-b)^2$ are used throughout. Nothing from the trigonometry, counting or sequences chapters is needed.

← 10.4 The Hyperbola 11.1 Coordinate Axes, Coordinate Planes and Octants →