The idea, step by step
Three coordinates, one location
A room needs three position numbers: across, depth and height. Once the axes are labelled, an ordered triple (x,y,z) specifies one position. Start at the origin, move x units in the x-direction, then y in the y-direction, then z in the z-direction. These are coordinates, not three separate points.
The order is fixed. In a warehouse, (3,2,4) could mean aisle position 3, depth 2 and shelf level 4. The point (3,4,2) swaps depth and height, so it is a different location. Always read the diagram’s axis labels rather than guessing what a slanted line represents.
Zero and negative coordinates
If z=0, a point is on the horizontal xy-plane: its height is zero. If x=0 or y=0, the point lies on the corresponding vertical coordinate plane. Two zero entries place a point on an axis; three zero entries locate the origin.
A negative coordinate tells you to move opposite that axis’s positive direction. It does not mean the whole triple is negative. For example, (−2,3,1) is 2 units in the negative x-direction, 3 in positive y and 1 in positive z. No vector multiplication is needed to locate it.
A common trap
A point with one zero coordinate must be the origin.
Instead: The origin is (0,0,0). One zero coordinate only places the point on a coordinate plane.
See the thinking
Read a shelf position
In a model warehouse, x is across, y is depth and z is shelf height. Describe P(3,2,4).
- Read the entries in x,y,z order.
- Move across 3 units, back 2 units and up 4 units.
- Do not exchange the last two entries.
A point on the floor
Which of A(2,1,0), B(0,2,3) and C(1,0,2) lies on the xy-plane?
- The xy-plane consists of points whose z-coordinate is zero.
- A has z=0. B and C have positive z-coordinates.
Check your understanding
What is the height of (4,2,5) if z measures height?
Show answer
5 units.
Height is the third coordinate, z.
Your turn
6 questions · 12 marks · Work at your own pace.
Read the questions online
- Read the x-, y- and z-coordinates of (2,5,1).2 marks
- A drone is 4 units across, 2 units back and 3 units high. Write its position as (x,y,z).2 marks
- Which is on the floor z=0: A(1,0,2), B(3,4,0) or C(0,2,1)?2 marks
- Describe (−2,1,3) using the positive axis directions as reference.2 marks
- Are (2,3,4) and (2,4,3) the same point? Explain.2 marks
- Which axis contains (0,0,5)?2 marks
Read the worked solutions
Equivalent correct methods earn the corresponding marks. Include units where asked.
- Read the x-, y- and z-coordinates of (2,5,1).2 marks
Read entries from first to third.
x=2, y=5, z=1. - A drone is 4 units across, 2 units back and 3 units high. Write its position as (x,y,z).2 marks
Across is x, depth is y and height is z.
(4,2,3) - Which is on the floor z=0: A(1,0,2), B(3,4,0) or C(0,2,1)?2 marks
B has third coordinate zero.
B. - Describe (−2,1,3) using the positive axis directions as reference.2 marks
The minus sign belongs only to x.
2 units in negative x, 1 in positive y and 3 in positive z. - Are (2,3,4) and (2,4,3) the same point? Explain.2 marks
The entries have a fixed order; swapping entries changes the location.
No. They have different y- and z-coordinates. - Which axis contains (0,0,5)?2 marks
Both horizontal entries are zero; only z varies.
The z-axis.
Curriculum reference & lesson scope
Australian Curriculum v9 and SA Curriculum Mathematics
Coordinate transformations review; identifying positions using three coordinates; interpreting 3D axes; describing routes and relative positions; comparing diagrams, views and digital representations. This is spatial location, not senior vector algebra.
Official curriculum source ↗