Mathematics 11–12 · Year 12

A vector walk: adding displacements on bearings

Introduction to vectors: Operating with vectors (Mathematics Extension 1, Year 12); Right-angled triangles (Mathematics Standard 1, Year 12); Trigonometry (Mathematics Standard 2, Year 12)

Practical, model not builtLow risk

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The idea

Two legs walked on bearings add as vectors, and the resultant displacement the learner tapes back to the start is the vector sum, not the sum of the leg lengths.

What you need

A sighting compass or a phone compass set to true north (or magnetic readings corrected by the local magnetic declination), a 100 m tape (or a pace count calibrated over 50 m), cones
An open field at least 130 m east-west by 120 m north-south (the two walking orders together span 126.6 m by 119.3 m)

How to do it

From a start cone walk 100 m on a true bearing of 060 degrees and place a cone; then walk 80 m on 150 degrees and place a second cone.
Tape the straight-line distance from the start to the second cone and read its bearing.
Resolve each leg into east and north components (d sin(bearing), d cos(bearing)) and add them.
Compute the magnitude and bearing of the resultant from its components; because the legs differ by 90 degrees, check the magnitude with Pythagoras as well.
Walk the legs in the opposite order and show that the finishing cone is in the same place.

What you should see

Legs of 100 m on 060 degrees and 80 m on 150 degrees have components (86.6, 50.0) m and (40.0, -69.3) m; the resultant (126.6, -19.3) m has magnitude 128.1 m on a bearing of 098.7 degrees, the same length Pythagoras gives from sqrt(100^2 + 80^2). The learner knows it worked when the taped distance is close to 128 m, far short of 180 m, and swapping the legs lands on the same point.

What changes

What you change: the two leg vectors (length and bearing)
What you measure: resultant displacement
What you keep the same: compass held level away from metal
pace length calibrated over 50 m

Common misconceptions

Each of these ideas is wrong, and the activity is a chance to test it.

Walking 100 m and then 80 m puts you 180 m from the start; only when both legs point the same way.
Bearings are measured anticlockwise from east like angles on axes; true bearings are measured clockwise from north.

Safety card

Low riskLearners carry it out

Hazards

uneven ground while reading a compass

Controls

look up between readings
check the field for holes first

Note

No chemicals and no heat are used, so the NSW Department of Education Chemical Safety in Schools package does not apply; outdoor work follows the school's grounds or excursion risk assessment (RiskAssess where the school uses it).

Curriculum references

The NSW syllabus outcomes and Australian Curriculum v9 codes this activity supports. They are references, not a verified or complete curriculum alignment.

Mathematics Extension 1 11–12 Syllabus (2024), Year 12 focus area Introduction to vectors; Year 11 taught from Term 1 2026, Year 12 from Term 4 2026, first HSC examination 2027 (the 2017 syllabus is still taught to Year 12 until then); page read 2026-09-22ME1-12-02
Mathematics Standard 11–12 Syllabus (2024), Year 12 Standard 1 focus area Right-angled triangles; Year 11 taught from Term 1 2026, Year 12 from Term 4 2026, first HSC examination 2027 (the 2017 syllabus is still taught to Year 12 until then); page read 2026-09-22MST-12-S1-04
Mathematics Standard 11–12 Syllabus (2024), Year 12 Standard 2 focus area Trigonometry; Year 11 taught from Term 1 2026, Year 12 from Term 4 2026, first HSC examination 2027 (the 2017 syllabus is still taught to Year 12 until then); page read 2026-09-22MST-12-S2-04
Australian Curriculum v9No Australian Curriculum v9 code is listed.

Sources

The pages the author read to write this activity.

All Concept Studio activities