To find the displacement vector between two points, you subtract the initial position vector from the final position vector.
Understanding Displacement
Displacement is a vector quantity that represents the change in an object's position. It points directly from the initial position to the final position and its magnitude is the straight-line distance between these two points. Unlike distance traveled, displacement only cares about the starting and ending locations, not the path taken.
Think of position vectors as arrows drawn from a fixed origin point (like the center of a coordinate system) to each point of interest. Let the initial position be point A and the final position be point B.
- The position vector of point A is r₀ (read as "r-naught" or "r-initial").
- The position vector of point B is rf (read as "r-final").
The displacement vector, often denoted as Δr (read as "delta r"), describes the change in position from A to B.
The Calculation Method
As indicated in the reference, the fundamental way to calculate the displacement vector is:
Displacement Vector = Final Position Vector - Initial Position Vector
In mathematical terms:
Δr = rf - r₀
Calculating with Coordinates
Position vectors are typically represented using coordinates in a coordinate system (e.g., 2D Cartesian (x, y) or 3D Cartesian (x, y, z)).
If your initial point A has coordinates (x₀, y₀, z₀), its position vector is r₀ = (x₀, y₀, z₀).
If your final point B has coordinates (xf, yf, zf), its position vector is rf = (xf, yf, zf).
To find the displacement vector Δr, you subtract the corresponding components:
Δr = (xf - x₀, yf - y₀, zf - z₀)
Or, if working in unit vector notation (i, j, k for x, y, z directions):
r₀ = x₀i + y₀j + z₀k
rf = xfi + yfj + zfk
Δr = (xf - x₀)i + (yf - y₀)j + (zf - z₀)k
Practical Example
Let's say an object moves from point A (2, 3) to point B (7, 5) in a 2D plane.
- Identify the initial position vector: The initial point is A(2, 3). So, r₀ = (2, 3).
- Identify the final position vector: The final point is B(7, 5). So, rf = (7, 5).
- Subtract the initial from the final:
Δr = rf - r₀
Δr = (7, 5) - (2, 3)
Δr = (7 - 2, 5 - 3)
Δr = (5, 2)
So, the displacement vector from point A to point B is (5, 2). This means the object's position changed by 5 units in the x-direction and 2 units in the y-direction.
Key Takeaway
Finding the displacement vector is a straightforward subtraction of vectors. Always remember to subtract the initial position vector from the final position vector to get the correct direction and magnitude of the displacement.
