Example Data Table
These sample cases help you verify your inputs and outputs.
| Method | Inputs | Expected |R| (approx.) | Expected direction (deg, approx.) |
|---|---|---|---|
| Magnitude + angle | A=10, B=7, θ=45° | ≈ 16.065 | ≈ 18.435° |
| Components | A=(5,3), B=(2,4) | ≈ 9.220 | ≈ 41.186° |
| Polar | A=8∠20°, B=6∠120° | ≈ 6.719 | ≈ 60.385° |
Formula Used
- Triangle law (two vectors with included angle θ):
|R| = √(A² + B² + 2AB cos θ)
- Component addition (always valid):
Rx = Ax + Bx, Ry = Ay + By|R| = √(Rx² + Ry²)
- Direction from +x axis:
angle(R) = atan2(Ry, Rx)
- Polar to components:
x = r cos α, y = r sin α
How to Use This Calculator
- Choose an input method that matches your data.
- Enter the vector values and optional unit label.
- Set decimals and angle units for the output display.
- Press Calculate to see results above the form.
- Use the download buttons to export CSV or PDF.
FAQs
1) What is the triangle law of vectors?
It adds two vectors by placing the tail of the second at the head of the first. The side from the first tail to the second head is the resultant.
2) When should I use magnitude and included angle?
Use it when you know both magnitudes and the angle between them. It’s common in physics problems using forces or displacements.
3) Why do components always work?
Components describe vectors on perpendicular axes. Adding x parts and y parts separately is equivalent to geometric addition in any direction.
4) What does atan2 do?
atan2(Ry, Rx) returns the correct direction angle for all quadrants. It avoids sign mistakes that happen with a simple Ry/Rx arctangent.
5) What is the equilibrant vector?
The equilibrant has the same magnitude as the resultant but points in the opposite direction. Adding it to the resultant gives zero, closing the triangle.
6) Can angles be negative or above 360°?
Yes. The calculator accepts them and computes using trigonometric functions. Degree outputs are wrapped into 0–360° for readability.
7) Why does my direction look different from a textbook?
Books may measure angles from a different axis or use clockwise conventions. This tool measures from +x and increases counterclockwise.
8) How can I verify my result quickly?
Compute Rx and Ry, then check |R| = √(Rx² + Ry²). The example table provides known cases to compare your outputs.