ACARA v9 CONTENT DESCRIPTION “describe the position and location of objects in 3 dimensions in different ways, including using a three-dimensional coordinate system with the use of dynamic geometric software and other digital tools”
From two dimensions to three
In two dimensions a point needs only two numbers. We give an across value and an up value, write them together as (x, y), and the pair pins the point to one spot on a flat sheet. The world we move through has depth as well as width and height, so two numbers are no longer enough to say where something sits. To capture that depth we add a third number and a third axis, usually called z, set at right angles to the flat x-y plane. With that one addition every pair (x, y) grows into a triple (x, y, z), and the new third value records how far the point rises above, or drops below, the flat plane we began with. The step from a sheet of paper to the room around it is exactly the step from two numbers to three.
From a plane to space
A third axis turns (x, y) into (x, y, z).
a third axis, z, added at right angles to the flat x-y plane lets a point be located in space as (x, y, z).
The three axes and the origin
Three axes carry the three numbers. The x-axis runs across, the y-axis runs back into the distance, and the z-axis runs straight up, and each one is set at right angles to the other two. They all begin at a single shared point called the origin, written (0, 0, 0), where x, y and z are every one of them zero. Each position in space is then measured from the origin: so far along x, so far along y, and so far up z. Because the axes are fixed and meet at one agreed starting point, two people reading the same triple will always arrive at the very same place.
Three axes meet at the origin
x across, y back, z up, all from (0, 0, 0).
three perpendicular axes x, y and z meet at the origin (0, 0, 0); x runs across, y runs back, and z runs up.
Reading an ordered triple
A point in space is written as an ordered triple (x, y, z), and the word ordered is the key to it. The first number is always the distance along x, the second is always the distance along y, and the third is always the height up z. To plot the triple (2, 3, 1) we start at the origin, step 2 along the x-axis, then step 3 in the y direction, and finally rise 1 up the z-axis, and the place we land is the point. Reading runs the other way just as smoothly: stand at a marked point, measure back to each axis in turn, and the three readings rebuild the triple. Holding the numbers in their fixed roles is what keeps the whole system dependable.
Plotting an ordered triple
Move x, then y, then z to reach the point.
to plot (2, 3, 1), go 2 along x, then 3 along y, then 1 up z, and mark the point.
Locating the corners of a box
A rectangular box, sitting square with the axes, has eight corners, and each corner is its own ordered triple. The four corners on the base all share the same height, z = 0, while the four corners on top all share the height of the box. If the base reaches 4 along x and 3 along y, its base corners are (0, 0, 0), (4, 0, 0), (4, 3, 0) and (0, 3, 0), and a box 2 high simply repeats those four with z raised to 2. Tracing one edge at a time shows a tidy pattern: along any single edge only one of the three coordinates changes while the other two hold still. Listing the eight triples is a clear, careful way to describe a solid shape using nothing but numbers.
The eight corners of a box
Each corner is an ordered triple.
the eight corners of a box are ordered triples; the base sits at z = 0 and the top at z = 2.
Coordinates in the real world
Three numbers locate real things, not only points in a diagram. A seat in a large arena can be found from its row, its seat along that row, and the tier it sits on, which is three values working together. A point inside a 3D model or a video game is held as a small block called a voxel, fixed by three coordinates. A place on Earth is pinned by its longitude, its latitude and its altitude above sea level. Different names sit on the front of each example, yet the same ordered-triple idea quietly runs underneath them all, which is why learning to read one triple helps you read every other.
Triples in the real world
A seat located by row, seat and tier.
real positions use triples too -- a seat by row, seat and tier, or a place by longitude, latitude and altitude.
Why this matters
Almost everything around us sits in three dimensions, so describing where a point is with three numbers is a foundation that later work leans on heavily. It underlies 3D modelling, computer games, engineering drawings, navigation and the mapping of the world. Two habits keep the work reliable: hold the axes in their agreed order, and read every coordinate from the origin rather than from some nearby point. The slip to watch for is swapping the y and the z value, or otherwise losing the order inside the triple, because a reordered triple usually names a different place. This unit stays with naming and locating positions; measuring the distance between two points in space, and working with vectors, build on these ideas in later years.
Quick self-check
1. How many numbers are needed to locate a point in three dimensions?
2. What are the coordinates of the origin in 3D?
3. In the point (2, 3, 1), which coordinate usually gives the height up the z-axis?
4. On a box whose base lies at z = 0, what is the z-coordinate of every base corner?
5. Why does the order of the numbers in a triple matter?