The idea, step by step
Read the labels before the perspective
A flat page can only show a perspective sketch of three perpendicular directions. The x-, y- and z-axes may look slanted even though they represent right angles in space. Their labels determine the coordinate order. Do not measure an apparent angle or length on this sketch to infer a coordinate.
The dashed path in the figure starts at the origin, reaches x=2, moves parallel to the y-axis until y=3, then moves parallel to the z-axis until z=2. It therefore locates Q(2,3,2). Numbered tick marks, rather than the apparent length of a line, give the units.
Use projection and checking
A construction line parallel to an axis changes only that coordinate. If a move is parallel to z, the x- and y-coordinates stay fixed. Dropping a point vertically to the xy-plane replaces its height by zero, giving a useful floor reference.
To check a plotted point, trace back to the floor and then to the axes. State any assumptions such as the axis scale. A sketch without labels, tick values or a stated scale cannot support an exact coordinate reading; request that information instead of inventing it.
A common trap
The sloping axis on the page must be z.
Instead: Perspective drawings can place axes in different directions. Read the axis labels; z is not identified by slant alone.
See the thinking
Read a construction
Use the labelled construction to state Q’s coordinates.
- The first leg reaches x=2.
- The second leg runs parallel to y and reaches y=3.
- The final vertical leg reaches z=2.
Move parallel to an axis
A point starts at (2,3,2). It moves parallel to the z-axis until its height is 5. Find its new position.
- Only the z-coordinate changes during a z-parallel move.
- Keep x=2 and y=3, and replace z=2 by z=5.
Check your understanding
Drop (3,4,2) vertically to the xy-plane. What point is reached?
Show answer
(3,4,0)
A vertical z-direction move keeps x and y unchanged and reduces the height to zero.
Your turn
6 questions · 12 marks · Work at your own pace.
Read the questions online
- State Q’s coordinates from the labelled diagram.2 marks
- From (1,2,3), move parallel to x until x=4. State the new position.2 marks
- Find the floor projection of (5,1,4).2 marks
- A point is on the y-axis with y=4. Write its coordinates.2 marks
- Can you read an exact point from an unlabelled 3D sketch with no scale? Explain.2 marks
- A construction reaches (3,0,0), then (3,2,0), then (3,2,4). Name the direction of each leg.2 marks
Read the worked solutions
Equivalent correct methods earn the corresponding marks. Include units where asked.
- State Q’s coordinates from the labelled diagram.2 marks
Read the x, then y, then z legs.
(2,3,2) - From (1,2,3), move parallel to x until x=4. State the new position.2 marks
Only the first entry changes.
(4,2,3) - Find the floor projection of (5,1,4).2 marks
Set z to zero, retaining x and y.
(5,1,0) - A point is on the y-axis with y=4. Write its coordinates.2 marks
On the y-axis, x and z are zero.
(0,4,0) - Can you read an exact point from an unlabelled 3D sketch with no scale? Explain.2 marks
A drawn length alone does not identify a coordinate value.
No. Axis identities and coordinate scales are missing. - A construction reaches (3,0,0), then (3,2,0), then (3,2,4). Name the direction of each leg.2 marks
Each leg changes one coordinate from the previous point.
Positive x, then positive y, then positive z.
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 ↗