Finding the displacement vector is a fundamental skill in physics and engineering that allows you to describe how an object’s position changes from one point to another. Even so, unlike distance, which only tells you how much ground has been covered, the displacement vector captures both the magnitude and the direction of the change in position, making it essential for analyzing motion, forces, and fields. In this guide you will learn the core concepts behind displacement, the step‑by‑step procedure to compute it, and practical examples that illustrate its application in one‑, two‑, and three‑dimensional spaces It's one of those things that adds up..
This changes depending on context. Keep that in mind.
Introduction to Vectors and Displacement
A vector is a quantity that possesses both magnitude and direction. Plus, the displacement vector, often denoted as (\vec{d}) or (\Delta \vec{r}), points from the initial position (\vec{r}_i) to the final position (\vec{r}_f). Here's the thing — common examples include velocity, acceleration, force, and, of course, displacement. Its magnitude equals the straight‑line distance between those two points, while its direction is the line that joins them, oriented from start to finish Practical, not theoretical..
Understanding how to find the displacement vector requires familiarity with coordinate systems, vector subtraction, and basic trigonometry. Whether you are working on a straight line, a plane, or a three‑dimensional space, the same principles apply: subtract the initial coordinates from the final coordinates to obtain the components of the displacement vector Turns out it matters..
Step‑by‑Step Procedure to Find the Displacement Vector
Below is a clear, repeatable method you can follow for any problem involving displacement.
1. Define the Coordinate System
Choose a reference frame that simplifies the problem. Cartesian coordinates (x, y, z) are the most common, but polar or cylindrical systems may be more convenient for circular motion. Clearly label the axes and note the units (meters, centimeters, etc.) And that's really what it comes down to..
2. Identify Initial and Final Position Vectors
Write down the position of the object at the start ((\vec{r}_i)) and at the end ((\vec{r}_f)) of the motion. Each position vector is expressed as a set of components: [ \vec{r}_i = \langle x_i,, y_i,, z_i \rangle \quad \text{or} \quad \vec{r}_f = \langle x_f,, y_f,, z_f \rangle ] If the motion occurs in only two dimensions, you can omit the (z) component.
3. Subtract the Initial Vector from the Final Vector
The displacement vector (\vec{d}) is obtained by vector subtraction: [ \vec{d} = \vec{r}_f - \vec{r}_i = \langle x_f - x_i,; y_f - y_i,; z_f - z_i \rangle ] This operation yields the components of (\vec{d}) along each axis.
4. Compute the Magnitude (if needed)
The magnitude of the displacement vector, often written as (|\vec{d}|) or (d), is found using the Pythagorean theorem: [ |\vec{d}| = \sqrt{(x_f - x_i)^2 + (y_f - y_i)^2 + (z_f - z_i)^2} ] For two‑dimensional cases, drop the (z) term Surprisingly effective..
5. Determine the Direction (if needed)
Direction can be expressed as an angle relative to a reference axis or as a unit vector And that's really what it comes down to..
- Unit vector: (\hat{d} = \dfrac{\vec{d}}{|\vec{d}|})
- Angle in 2D: (\theta = \tan^{-1}!\left(\dfrac{y_f - y_i}{x_f - x_i}\right)) (adjust quadrant as necessary)
- Direction cosines in 3D: (\cos\alpha = \dfrac{x_f - x_i}{|\vec{d}|}), (\cos\beta = \dfrac{y_f - y_i}{|\vec{d}|}), (\cos\gamma = \dfrac{z_f - z_i}{|\vec{d}|})
6. State the Result Clearly
Present the displacement vector in component form, and optionally include its magnitude and direction. For example: [ \vec{d} = \langle 3,\text{m},; -4,\text{m},; 0,\text{m} \rangle,\quad |\vec{d}| = 5,\text{m},\quad \theta = -53.1^\circ \text{ from the +x axis} ]
Mathematical Formulation Summary
| Step | Operation | Formula |
|---|---|---|
| 1 | Choose axes | – |
| 2 | Write position vectors | (\vec{r}_i = \langle x_i, y_i, z_i \rangle), (\vec{r}_f = \langle x_f, y_f, z_f \rangle) |
| 3 | Subtract | (\vec{d} = \langle x_f-x_i, y_f-y_i, z_f-z_i \rangle) |
| 4 | Magnitude | ( |
| 5 | Direction | Unit vector (\hat{d} = \vec{d}/ |
| 6 | Report | (\vec{d}), ( |
Worked Examples
Example 1: One‑Dimensional Motion
A particle moves from (x_i = 2,\text{m}) to (x_f = 7,\text{m}) along the x‑axis Still holds up..
- (\vec{d} = \langle 7-2 \rangle = \langle 5 \rangle,\text{m})
- Magnitude: (|\vec{d}| = 5,\text{m})
- Direction: positive x‑direction.
Example 2: Two‑Dimensional Motion
A drone flies from point (A(1,,3)) to point (B(-4,,6)) (coordinates in meters).
- (\Delta x = -4 - 1 = -5)
- (\Delta y = 6 - 3 = 3)
- (\vec{d} = \langle -5,, 3 \rangle,\text{m})
- Magnitude: (|\vec{d}| = \sqrt{(-5)^2 + 3^2} = \sqrt{34} \approx 5.83,\text{m})
- Angle: (\theta = \tan^{-1}(3/ -5) \approx 149.0^\circ) measured from the +x axis (since the vector lies in the second quadrant).
Example 3: Three‑Dimensional Motion
A submarine starts at ((0,,0,,-200),\text{m}) and ends at ((15
