Enter values and click “Calculate displacement and angle” to see detailed results here, including multiple angle conventions and azimuth.
| Case | Mode | x₁ | y₁ | x₂ | y₂ | Δx | Δy | |s| | Angle (°) | Angle (rad) | Azimuth (°) | Quadrant | Unit |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Example | Coordinates | 0 | 0 | 3 | 4 | 3 | 4 | 5 | 53.1301 | 0.9273 | 36.8699 | First quadrant | m |
Displacement describes the straight-line change in position from an initial point (x₁, y₁) to a final point (x₂, y₂). The horizontal and vertical components of displacement are:
- Δx = x₂ − x₁
- Δy = y₂ − y₁
In components mode, Δx and Δy are entered directly and treated as the vector’s components from the origin.
The magnitude of the displacement vector, |s|, is obtained using the Pythagorean theorem:
|s| = √(Δx² + Δy²)
The displacement angle θ is measured from the positive x-axis, counterclockwise, using:
θ = atan2(Δy, Δx)
The calculator converts this value into both 0°–360° and −180°–180° conventions, and also reports the azimuth measured from North, clockwise.
- Select Coordinates if you know initial and final positions.
- Select Components if you know Δx and Δy directly.
- Choose the displacement unit that matches your problem statement.
- Select the angle display mode and preferred decimal precision.
- Enter values, then click “Calculate displacement and angle”.
- Review magnitude, angle in both conventions, and azimuth from North.
- Use Download CSV or Download PDF to save or share results.
The following example cases illustrate how different coordinate or component sets influence displacement magnitude, angle, and quadrant classification.
| Case | Mode | x₁ | y₁ | x₂ | y₂ | Δx | Δy | |s| | Angle (°) | Azimuth (°) | Quadrant |
|---|---|---|---|---|---|---|---|---|---|---|---|
| A | Coordinates | 0 | 0 | 5 | 0 | 5 | 0 | 5 | 0 | 90 | Along horizontal axis |
| B | Coordinates | 0 | 0 | 0 | -6 | 0 | -6 | 6 | 270 | 180 | Along vertical axis |
| C | Components | 0 | 0 | 3 | 4 | 3 | 4 | 5 | 53.13 | 36.87 | First quadrant |
| D | Components | 0 | 0 | -2 | -5 | -2 | -5 | ≈ 5.39 | ≈ 248.20 | ≈ 201.80 | Third quadrant |
1. Understanding displacement angle
The displacement angle describes the direction of a displacement vector relative to the positive x-axis. Together with magnitude, it converts raw coordinate differences into a meaningful description of how far and in which direction motion occurs in a two-dimensional plane.
2. Coordinates versus components
Sometimes problems specify starting and ending coordinates, while others give horizontal and vertical components directly. This calculator supports both representations through the input mode selector so you can use whichever description your textbook, experiment, or simulation already provides.
When you choose coordinate input, the tool automatically computes Δx and Δy. When you choose component input, it treats those values as the vector itself starting from the origin.
3. Computing magnitude of displacement
The magnitude |s| is found using the Pythagorean theorem applied to Δx and Δy. This gives the straight-line distance between initial and final positions, independent of any curved path the object may actually have followed in between.
This distance can be expressed in meters, centimeters, kilometers, feet, or miles. Selecting the unit that matches your data keeps intermediate calculations and final results clearly documented and easy to compare across different cases.
4. Angle conventions used
Different fields adopt different angle ranges. Many engineers use 0°–360° measured counterclockwise from the positive x-axis. Other references prefer −180°–180° so that directions to the left of the x-axis appear as negative angles.
The calculator reports both conventions simultaneously. You can also select which one appears as the primary display angle, making it easier to match your lecture notes, examination marking schemes, or reference software.
5. Azimuth from North
Surveying, navigation, and mapping frequently rely on azimuth, measured clockwise from geographic North. By converting the displacement direction into azimuth, this calculator connects traditional vector analysis with map-based descriptions used in fieldwork and geospatial applications.
This is especially helpful when translating coordinate-based motion into compass bearings or when checking the consistency of GPS tracks, drone flight logs, or outdoor experiment results.
6. Educational and self-study benefits
Students can experiment with different coordinate pairs to see how magnitude and angle respond. Visualizing quadrant changes and axis-aligned motion builds intuition for trigonometry, vector decomposition, and kinematic diagrams used in introductory physics courses.
7. Data export and documentation
The built-in CSV export sends the results table directly into spreadsheet software for further analysis or graphing. The print-ready PDF style output simplifies attaching clean numerical tables to laboratory reports, project documentation, or homework submissions.
Organized records of displacement angles support clear communication, easier grading, and reliable long-term archives of engineering or scientific work.
What does the displacement angle represent?
It represents the direction of the displacement vector relative to the positive x-axis. Combined with the magnitude, it tells you how far and in which direction an object has moved between two positions in a two-dimensional plane.
When should I use coordinates instead of components?
Use coordinates when you know initial and final positions from graphs, maps, or experiments. Use components when you already have horizontal and vertical changes, for example from algebraic solutions, force diagrams, or output generated by simulation software.
Why are there two different angle conventions?
Some courses and software use 0°–360°, while others prefer −180°–180°. Showing both makes it easier to compare results, switch between sign conventions, and avoid confusion when checking answers against textbooks, lecture notes, or published example solutions.
What is the purpose of azimuth from North?
Azimuth is useful when relating vector directions to maps or navigation bearings. It measures the direction clockwise from geographic North, so surveyors, navigators, and field scientists can interpret displacement results like compass headings rather than pure mathematical angles.
Can this calculator handle negative coordinates?
Yes. Negative coordinates and components are handled naturally through the underlying trigonometric functions. The calculator automatically places the displacement in the correct quadrant and adjusts angle values so your results remain consistent with standard vector analysis rules.
How can I save or share my calculations?
You can download the results table as a CSV file and open it in spreadsheet software, or use the print to PDF option to create clean, shareable pages for homework, reports, or project documentation.