The idea, step by step
Record the position after each instruction
For a route in three dimensions, write a new triple after every move. Moving 3 units in the positive x-direction adds 3 to x; moving 2 units in negative y subtracts 2 from y. The other entries stay unchanged. The starting triple is essential: a route does not have a unique final point without it.
For an axis-aligned route, total distance travelled is the sum of the positive lengths of all the legs, even when a later move reverses direction. A change from x=1 to x=5 and back to x=3 travels 6 units along x, although the final x-coordinate is only 2 greater than at the start.
Compare locations and plan a route
To move from A to B using the coordinate directions, compare corresponding entries. If y decreases from 4 to 1, travel 3 units in negative y. One valid route changes x, then y, then z; changing the order gives a different path with the same endpoint.
The total axis-aligned length is the sum of those three coordinate changes taken as non-negative distances. It is not generally the straight-line distance between the points. State whether a problem asks for the actual route, an axis-aligned route, or just the destination.
A common trap
A negative movement subtracts distance from the total journey.
Instead: It decreases a coordinate, but distance travelled is always added as a non-negative length.
See the thinking
Follow a route
Start at (1,2,0). Move 3 units in positive x, 2 units in positive z and 1 unit in negative y. Give the endpoint and route length.
- After the first move: (4,2,0).
- After the second: (4,2,2).
- After the last: (4,1,2).
- Add leg lengths: 3+2+1=6.
Plan between two points
Give an axis-aligned route from A(1,4,2) to B(3,1,5), and its length.
- Increase x by 2.
- Decrease y by 3.
- Increase z by 3.
- The route length is 2+3+3=8.
Check your understanding
Start at (2,1,3) and move 4 units in negative x. What is the endpoint?
Show answer
(−2,1,3)
Subtract 4 from x only: 2−4=−2.
Your turn
6 questions · 12 marks · Work at your own pace.
Read the questions online
- Start at (0,0,0). Move 2 in positive x, 3 in positive y and 1 in positive z. Give the endpoint.2 marks
- Start at (3,2,1). Move 2 in negative y. Give the endpoint.2 marks
- Start at (1,1,1). Move 4 in positive x then 2 in negative x. Give endpoint and distance.2 marks
- Describe the change from A(2,5,1) to B(2,1,4).2 marks
- Find the length of an axis-aligned route from (1,2,3) to (4,1,5) without reversing direction.2 marks
- Mia says returning to the start means the total distance is zero. She walks 3 units out and 3 back. Correct her.2 marks
Read the worked solutions
Equivalent correct methods earn the corresponding marks. Include units where asked.
- Start at (0,0,0). Move 2 in positive x, 3 in positive y and 1 in positive z. Give the endpoint.2 marks
Apply each change to its matching entry.
(2,3,1) - Start at (3,2,1). Move 2 in negative y. Give the endpoint.2 marks
2−2=0 in the y-entry.
(3,0,1) - Start at (1,1,1). Move 4 in positive x then 2 in negative x. Give endpoint and distance.2 marks
Final x=1+4−2=3. Distance=4+2=6.
(3,1,1); 6 units. - Describe the change from A(2,5,1) to B(2,1,4).2 marks
Compare corresponding entries: Δx=0, Δy=−4, Δz=3.
4 units in negative y and 3 units in positive z; x stays unchanged. - Find the length of an axis-aligned route from (1,2,3) to (4,1,5) without reversing direction.2 marks
The leg lengths are 3, 1 and 2. Add: 3+1+2=6.
6 units. - Mia says returning to the start means the total distance is zero. She walks 3 units out and 3 back. Correct her.2 marks
Both legs contribute distance: 3+3=6.
Her final position change is zero, but she travels 6 units.
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 ↗