Describing a Change of Location
A location in three-dimensional space is described by three coordinates. When an object moves from one location to another, its displacement records how each coordinate changes. This is useful for describing motion even when the route includes several turns or changes in height.
The order of the two locations matters. Displacement is measured from a starting point to a finishing point, so reversing the trip reverses the signs of all three components. Each sign is interpreted relative to the chosen positive direction of its coordinate axis.
The components have the same units as the coordinates. For example, if the coordinates are measured in metres, each displacement component is measured in metres. A negative component means that the coordinate decreased; it does not mean that the object is in a negative or invalid location.
Why the Components Are Final Minus Initial
Each coordinate tells how far along one axis a point lies, with the sign indicating its side of the origin. Along the \(x\)-axis, the directed change from \(x_1\) to \(x_2\) is \(x_2-x_1\): adding that change to the starting coordinate must give the finishing coordinate. The same reasoning applies independently to the \(y\)- and \(z\)-coordinates.
Thus the three coordinate changes together describe exactly how the starting location becomes the finishing location. This also gives a direct check: add each displacement component to the matching starting coordinate. The resulting three coordinates must be those of the endpoint.
Worked Examples
Worked Example: Displacement Between Two Locations
A hiker starts at \(A=(2,-1,4)\) metres and finishes at \(B=(7,3,1)\) metres. Find the displacement and interpret its components, taking positive \(x\) as east, positive \(y\) as north, and positive \(z\) as up.
The initial coordinates are \((2,-1,4)\), and the final coordinates are \((7,3,1)\). Subtract the initial coordinate from the final coordinate in each position:
Therefore, \(\mathbf d_{AB}=(5,4,-3)\) metres. The hiker's \(x\)-coordinate increases by \(5\) metres, so the \(x\)-change is eastward. The \(y\)-coordinate increases by \(4\) metres, so the \(y\)-change is northward. The \(z\)-coordinate decreases by \(3\) metres, so the \(z\)-change is downward.
Check by applying the components to the starting coordinates:
This reproduces \(B\), so the displacement is consistent with the stated endpoints.
Worked Example: Finding Overall Displacement Along a Route
A cart moves through the locations \(P=(0,0,0)\), \(A=(4,0,0)\), \(B=(4,-3,2)\), and \(C=(1,-3,2)\), with coordinates in metres. Assume positive \(x\) is east, positive \(y\) is north, and positive \(z\) is up. Find the overall displacement from \(P\) to \(C\), and describe the coordinate changes along the route.
First, record the change in each coordinate at each leg. From \(P\) to \(A\), the changes are \((4-0,\;0-0,\;0-0)=(4,0,0)\) metres. From \(A\) to \(B\), they are \((4-4,\;-3-0,\;2-0)=(0,-3,2)\) metres. From \(B\) to \(C\), they are \((1-4,\;-3-(-3),\;2-2)=(-3,0,0)\) metres.
For the overall displacement, use the beginning and ending locations, not just the last leg:
The cart ends \(1\) metre farther in the positive \(x\)-direction, \(3\) metres farther in the negative \(y\)-direction, and \(2\) metres higher than it started. With the stated axis directions, these are eastward, southward, and upward coordinate changes, respectively.
As a check, start at \(P\) and apply the overall coordinate changes: \((0+1,\;0+(-3),\;0+2)=(1,-3,2)=C\). The endpoint is recovered.
The route description and overall displacement answer different questions. The route has three legs, including a \(4\)-metre eastward change and then a \(3\)-metre westward change in the \(x\)-coordinate. Those changes partly offset, leaving an overall \(x\)-change of \(1\) metre. The displacement records only the net change in location; it does not describe every turn or the full path travelled.
Worked Example: Interpreting Negative Components
A remotely operated vehicle moves from \(R=(12,5,8)\) kilometres to \(S=(4,9,2)\) kilometres. Let positive \(x\) point east, positive \(y\) point north, and positive \(z\) point upward. Find its displacement and explain each sign.
Subtract the coordinates of \(R\) from those of \(S\), one coordinate at a time:
Hence \(\mathbf d_{RS}=(-8,4,-6)\) kilometres. The negative \(x\)-component means the vehicle's \(x\)-coordinate decreased by \(8\) kilometres, or changed westward. The positive \(y\)-component means a \(4\)-kilometre northward change. The negative \(z\)-component means a \(6\)-kilometre downward change.
Check by applying the components to \(R\): \((12+(-8),\;5+4,\;8+(-6))=(4,9,2)=S\). This confirms the signs and values.
Reading Signs in Context
A sign has meaning only in relation to the chosen positive direction of its axis. If positive \(y\) is north, a negative \(y\)-component describes a southward coordinate change. If the positive direction is defined differently, the verbal interpretation changes too. The subtraction rule does not change: the final coordinate minus the initial coordinate is still the component.
It is also important to distinguish a coordinate from a coordinate change. A point can have a negative coordinate, such as \(y=-1\), while an object's change in \(y\) can be positive or negative. For example, moving from \(y=-1\) to \(y=3\) gives \(3-(-1)=4\), a positive change, even though the starting coordinate was negative.
| Component sign | Coordinate change | Interpretation when positive axes are east, north, and up |
|---|---|---|
| Positive | The coordinate increased | Change in the positive axis direction |
| Negative | The coordinate decreased | Change opposite the positive axis direction |
| Zero | The coordinate stayed the same | No change along that coordinate axis |
A Common Subtraction Error
A frequent mistake is to subtract the final point from the initial point even though the motion is described from the initial point to the final point. That calculation answers the reverse-direction question.
For the hiker's trip from \(A=(2,-1,4)\) to \(B=(7,3,1)\), reversing the subtraction gives \((2-7,\;-1-3,\;4-1)=(-5,-4,3)\), not the displacement from \(A\) to \(B\). Applying this incorrect result to \(A\) gives \((2-5,\;-1-4,\;4+3)=(-3,-5,7)\), which is not \(B\).
The correction is to follow the stated direction: endpoint minus starting point. Using \(B-A\) gives \((5,4,-3)\), and the check \((2+5,\;-1+4,\;4-3)=(7,3,1)\) recovers \(B\). If the direction of travel were instead from \(B\) to \(A\), then \((-5,-4,3)\) would be the correct displacement.
Why Displacement Is Useful
Displacement gives a compact description of how a position changes. It allows you to report the net coordinate change without listing every intermediate location. This is useful in route planning, navigation, surveying, and motion problems where the starting and finishing positions matter.
However, displacement alone does not specify the route. Two trips can start and finish at the same locations and therefore have the same displacement, even if their routes differ. When a problem asks for displacement, focus on the change from the initial location to the final location. When it asks about the route, intermediate locations and turns may also be needed.
Check Your Understanding
For each question, keep the order of travel explicit and use the coordinate signs relative to the stated axes.
- If an object moves from \((3,-2,1)\) to \((8,1,-4)\), what is its displacement vector?
- An object moves from \(y=6\) to \(y=-2\). What is its \(y\)-displacement, and what does its sign tell you?
- A route starts at \((0,0,0)\), passes through \((2,1,0)\), and ends at \((2,1,5)\). Find the overall displacement from start to finish.
- Why does a displacement vector not, by itself, specify the full route travelled?
- What endpoint should result when the displacement components are applied to the starting coordinates?