Lesson 5 of 6 · Components, displacement & reference frames
Keep direction in the calculation
8 minute lesson · 3 checks along the way
Your goal for this lessonAdd displacement components in a defined local frame and distinguish net displacement, path length, and coordinate choices.
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A direction is part of the quantity
A distance of ten meters does not tell a rover where to go. A includes both magnitude and direction: it connects a starting position to an ending position. We can represent it with an arrow or with along defined axes. A scalar, such as elapsed time or path length, does not require a spatial direction. Velocity is a vector; speed is its nonnegative magnitude.
Define a small, flat practice map with the lander as origin, positive x pointing east, and positive y pointing north. Positions and displacements use meters. The coordinate pair (2, −1) means two meters east and one meter south of the origin. Negative signs specify direction relative to the chosen axes; they do not mean an impossible negative amount of physical space.
Add components, then find the magnitude
Our fictional rover starts at (2, −1), drives six meters east, and then eight meters north. Its displacement is (6, 8), and its final position is (2 + 6, −1 + 8) = (8, 7). Add each component to the matching component. Adding eastward meters to northward meters as if both were the same direction loses information that the route planner needs.
The two straight legs have total path length 6 + 8 = 14 meters. The net displacement has magnitude √(6² + 8²) = 10 meters, using the right triangle between start and finish. These answers do not conflict: one follows the route; the other measures the direct separation of its endpoints. The direct line may cross an obstacle, so the smaller number is not automatically a driveable shortcut.
Same endpoints, different length questions
Original fictional map with a common linear scale, defined east/north axes, and a lander origin. The dashed displacement does not assert a safe shortcut.
Open full-size diagram in a new tab: Same endpoints, different length questionsCheck 1 of 3
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Work through a decision
Repair an incomplete navigation instruction
A command says only “move ten meters,” while the target lies six meters east and eight meters north of the start. The distance alone is insufficient. Supply the or a magnitude with an unambiguous direction. Using angles counterclockwise from east, the direction is about 53.1 degrees because tan θ = 8/6. That is not the same convention as a compass bearing measured clockwise from north.
Now redefine positive x to point west while leaving positive y north. The same physical displacement has components (−6, 8); its magnitude remains ten meters. Moving the origin also changes position coordinates, but a common translation of start and finish cancels when their difference is taken. Never combine coordinate pairs from different frames without first expressing them in the same frame.
Check 2 of 3
10 pointsUse the reading above. You can try again without losing points.
Connect the map to a real instrument
JPL describes rover navigation that combines camera information with planning and motion estimates. Image features can help estimate how a rover moved, but an image pixel and a map meter are different quantities. Converting between them needs camera geometry, range information, and an agreed frame. The diagram here supplies an invented flat route so you can practice the arithmetic without treating a photograph as a surveyed map.
On a whole planet, curvature and coordinate conventions require more care. Latitude and longitude are angular coordinates, and equal longitude differences do not correspond to equal surface distances at every latitude. Near a spacecraft, axes may instead follow its body or an orbital direction. Start every vector problem by naming the frame, units, and time. Components are descriptions of a physical relationship, not numbers with meaning independent of their coordinate system.
Pathfinder’s PIA01122 observation shows Sojourner near rocks called The Dice on mission sol 22. NASA combined observations through red, green, and blue filters into this color view. The visible tracks give route context, but the fictional 6-meter/8-meter exercise is not a measurement of these tracks.
Look closely
Which map information would you need before turning the visible tracks into east/north displacement components?
See what to notice
You need a physical scale, orientation, chosen origin, identifiable endpoints, and the camera-to-ground geometry. The image documents terrain and tracks; a numerical displacement cannot be read from screen pixels alone.
Check 3 of 3
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Put it together
Your lesson progress
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Go to the source
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