SCIENCE · INVESTIGATE AND EXPLAIN

Close a two-leg survey without confusing distance and displacement

A two-leg route can cover substantial ground while ending near its starting point. Distance adds lengths of the traveled legs. Displacement connects the starting and ending positions and carries direction. Use one shared east–north coordinate system to combine the two legs; do not add their magnitudes unless their directions justify that special case.

Learning outcome: Hold second-leg length 4 m and compare headings 0°,90°,180°. Retain components, path distance and the undefined direction at cancellation. Then compare lengths 3 and 4 m near 180° and discuss sensitivity rather than claiming high directional precision.

Starting recap

Use SI units, signed components, algebraic ratios and graph coordinates. Identify the measurement rule and distinguish a record from an interpolated model.

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Try

With a 4 m east leg followed by 3 m at 120° from east, components are (2.5,2.59808) m. Resultant magnitude is √13≈3.60555 m, while total path is 7 m.

Open the experiment

Predict

Can total distance increase while the resultant displacement gets smaller?

Your prediction and reason

Experiment

Hold second-leg length 4 m and compare headings 0°,90°,180°. Retain components, path distance and the undefined direction at cancellation. Then compare lengths 3 and 4 m near 180° and discuss sensitivity rather than claiming high directional precision.

Resolve a survey displacement and a cancellation

Worked teaching model · synthetic values · First leg is4 m east, second has declared length/heading in a flat plane. Components use common east/north axes. Path length is sum of leg lengths; zero resultant has no direction. Not a navigation or field-safety instruction.

Second heading counterclockwise from east: 120 degrees Second leg length: 3 m

Reset keeps your prediction, captures and practice history.

Survey legs and resultant on equal spatial scales

Spatial axes use the same pixels per physical unit, preserving geometry. 1: two-leg path (solid).2: resultant displacement (dashed). Joined curves show the supplied model, not additional observations.

Equivalent numerical coordinates for Survey legs and resultant on equal spatial scales

two-leg path

east coordinate (m)	north coordinate (m)
0	0
4	0
2.5	2.59808

At most 82 representative computed coordinates including endpoints; the plotted curve can use a finer grid. A table of a revealed generator is not independent measurement evidence.

resultant displacement

east coordinate (m)	north coordinate (m)
0	0
2.5	2.59808

At most 82 representative computed coordinates including endpoints; the plotted curve can use a finer grid. A table of a revealed generator is not independent measurement evidence.

Calculated quantities under the declared measurement model
Quantity	Value	Unit
East displacement	2.5	m
North displacement	2.59808	m
Resultant magnitude	3.60555	m
Path distance	7	m
Direction counterclockwise from east	46.1021	degrees

Components add in a shared frame; path distance and resultant magnitude answer different questions.

Heading is measured from east counterclockwise, not from north clockwise. A zero-length second leg makes its heading irrelevant. Near cancellation, a small component error can cause a large direction change; a direction rounded to a neat angle is not evidence of precision.

Predict before inspecting the values. Save contrasting records with assumptions and one alternative explanation. This virtual display does not certify practical measurement technique.

Components add in a shared frame; path distance and resultant magnitude answer different questions.

Observation to attach to a capture

Captures preserve recipes and replay their calculated results. They are synthetic model evidence, not physical measurements.

Understand

Begin with a measurement question

Build the algebraic model

The first leg is 4 m east. A second leg of length L at angle θ counterclockwise from east has components Lcosθ east and Lsinθ north. The total components are (4+Lcosθ,Lsinθ). Magnitude follows the Pythagorean theorem; direction must use the signs of both components. A plain inverse tangent of y/x can choose the wrong quadrant when x is negative, so interpret the geometry or use a two-argument angle calculation.

Check a concrete implication

At θ=90° and L=3 m, the resultant is (4,3) m, with magnitude 5 m and direction about 36.87° north of east. The path distance is 7 m. At θ=180° and L=4 m, displacement is exactly zero while path distance is 8 m. Zero displacement has no direction; assigning 0° would falsely describe an eastward result.

Challenge the evidence boundary

The plotted path and resultant use equal spatial scales, so geometric direction is preserved. The numerical coordinate table remains the authoritative equivalent representation. Near cancellation, small component errors can rotate the inferred direction greatly even when the absolute displacement error is small. A neat angle is not evidence that a survey is precise. These supplied legs are not a physical navigation or field-safety recommendation.

Worked original case

With a 4 m east leg followed by 3 m at 120° from east, components are (2.5,2.59808) m. Resultant magnitude is √13≈3.60555 m, while total path is 7 m.

Predict and investigate

Can total distance increase while the resultant displacement gets smaller? Hold second-leg length 4 m and compare headings 0°,90°,180°. Retain components, path distance and the undefined direction at cancellation. Then compare lengths 3 and 4 m near 180° and discuss sensitivity rather than claiming high directional precision.

Changed numerical practice

Try the changed case before feedback

A route travels 6 m east then 8 m north. Find resultant magnitude in metres, and distinguish it from path distance.

Numerical result in the requested units

This practice is public and unsaved. A matching number does not verify understanding. Retain your explanation, assumptions and changed-case evidence in the separate course workspace.

Reflect and source-check

Why must the angle reference be stated before resolving components?

Original explanations, synthetic records and models; checked 30 October 2026 against OpenStax College Physics 2e. The guided calculation is algebra-based; no calculus prerequisite is assumed. All activity is virtual. No physical launch, drop, water crossing, machinery operation or medical decision is requested. A public answer or saved capture does not verify independent mastery or actual measurement technique.

Apply

Public practice checks a specific claim. It does not establish independent mastery.

Why must the angle reference be stated before resolving components?

Your explanation and limitations

Continue

Completion requires a saved prediction, at least one capture and a reflection. Presence checks do not grade your reasoning.

Skipping retains experiments and awards no mastery.

Previous: Calibrate position before interpreting a speed estimate Next: Find what a straight sampled track does not reveal

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