Calculate the resultant force vector using the parallelogram law of forces.
Enter the magnitudes of two forces \( F_1 \) and \( F_2 \) and the angle \( \theta \) between them.
* All values should be in SI units (Newtons for forces and degrees for the angle).
Step 1: Enter Parameters
Example: 10 N
Example: 15 N
Example: 60°
Physics Tutorial: Using a Resultant Force Vector Calculator
A resultant force is the single vector that has the same effect as two or more individual forces acting on a body. In mechanics, finding this vector is essential for predicting acceleration through Newton’s second law, \(\mathbf F_\text{net} = m\mathbf a\). This tutorial reviews vector addition, shows formulas for two‑ and three‑dimensional cases, and explains how to use the online resultant‑force calculator.
What Is a Resultant Force and Why Does It Matter?
- Definition: The vector sum of all forces on a body: \(\mathbf F_\text{R} = \displaystyle\sum_{i=1}^{n} \mathbf F_i\).
- Physical Meaning: Determines net acceleration and the motion of the object via \(m\mathbf a\).
- Applications: Statics (equilibrium checks), dynamics (projectile motion, vehicle traction), engineering design, robotics, and biomechanics.
- Big Idea: Multiple forces can always be replaced by a single equivalent vector acting at the same point.
Vector Addition Basics
Vectors add head‑to‑tail (triangle method) or via the parallelogram rule. In rectangular coordinates:
The components add separately:
How to Calculate Magnitude and Direction
2‑D Resultant
Given \(F_{Rx}\) and \(F_{Ry}\):
Angle \(\phi\) is measured from the +x axis; adjust for quadrant as needed.
3‑D Resultant
Add the z‑components and extend Pythagoras:
Direction cosines: \(\cos\alpha = F_{Rx}/F_R\), \(\cos\beta = F_{Ry}/F_R\), \(\cos\gamma = F_{Rz}/F_R\).
How to Use the Online Resultant Force Vector Calculator
- Open Resultant Force Calculator.
- Choose 2‑D or 3‑D mode.
- Enter the magnitude and direction (angle or components) of each force.
- Click Add Vector to include additional forces.
- Press Calculate Resultant to display magnitude, direction, and component breakdown.
Example Problems
Example 1 — Two Forces in 2‑D
F₁ = 50 N at 30°; F₂ = 70 N at 120°.
Components:
\(F_{1x}=43.3\;\text N,\; F_{1y}=25.0\;\text N\)
\(F_{2x}=-35.0\;\text N,\; F_{2y}=60.6\;\text N\)
Resultant:
\(F_{Rx}=8.3\;\text N,\; F_{Ry}=85.6\;\text N\)
\(F_R = \sqrt{8.3^2+85.6^2}\approx85.9\;\text N\),
\(\phi=\tan^{-1}(85.6/8.3)\approx84.5°\).
Example 2 — Three Forces in 3‑D
Given:
\(\mathbf F_1=\langle 10,\,5,\,0\rangle\;\text N\),
\(\mathbf F_2=\langle -4,\,12,\,-3\rangle\;\text N\),
\(\mathbf F_3=\langle 0,\,-7,\,9\rangle\;\text N\).
Sum: \(\mathbf F_R=\langle6,\,10,\,6\rangle\;\text N\).
Magnitude: \(F_R=\sqrt{6^2+10^2+6^2}\approx13.2\;\text N\).
Direction cosines:
\(\cos\alpha=0.455\), \(\cos\beta=0.758\), \(\cos\gamma=0.455\).
Frequently Asked Questions
What’s the difference between scalar and vector addition?
Scalars add arithmetically; vectors require component‑wise or graphical addition because they have direction as well as magnitude.
Can I mix degrees and radians?
Use one unit consistently. The calculator lets you choose degrees or radians and converts internally.
Does the order of addition matter?
No—vector addition is commutative: \(\mathbf A+\mathbf B=\mathbf B+\mathbf A\).
How does the calculator handle 3‑D forces?
It sums x‑, y‑, and z‑components separately, finds the magnitude with \(F_R=\sqrt{F_{Rx}^2+F_{Ry}^2+F_{Rz}^2}\), and reports direction cosines or azimuth/elevation angles.