Add up to five vectors and calculate the resultant in real time. Intuitively learn vector composition from three perspectives: origin display, tip-to-tail method, and component bar chart.
Presets
Vector List
Results
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Resultant |F|
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Direction θ
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Fx (total)
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Fy (total)
Origin
All vectors start from the same origin. The thick dashed arrow is the resultant vector.
Tip
Tip-to-tail method: connect each vector's tail to the previous vector's tip. The arrow from the first start point to the final endpoint is the resultant.
Components
Each vector's x-component (blue) and y-component (orange). The rightmost bars show the resultant components.
I know vectors are "quantities with magnitude and direction" from the textbook, but how do you add them? You can't just add them like ordinary numbers, right?
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Right. For example, a 3 N force to the east and a 4 N force to the north do not simply become 7 N. Add the x- and y-components separately: east gives x=3 and north gives y=4, so the resultant is √(3²+4²)=5 N at about 53° above east. It is the same geometry as finding the hypotenuse of a right triangle.
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I see! But what about vectors at an angle? When I see something like 30°, I can't immediately figure out the components...
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That is where sine and cosine come in. For a vector with magnitude F and angle θ, Fx = Fcosθ and Fy = Fsinθ. If F=10 N and θ=30°, then Fx≈8.66 N and Fy=5 N. After that, add the x-components and y-components separately. Try setting F1 to 10 N at 30° and look at the component graph.
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What is the tip-to-tail method? When I looked at the tip-to-tail tab, I saw a polyline, but I can't believe that's the same as vector addition...
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It can feel odd at first. Think of it as a sequence of moves: start at point A, move by vector F1, then from that new point move by F2, and so on. The arrow from the original start point to the final point is the resultant. Its magnitude and direction exactly match the component calculation, and the method is useful for structural force diagrams.
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I tried the "3-force equilibrium" preset and the resultant force became zero. What does it mean when the resultant force is zero?
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If the resultant force on an object is zero, the object remains at rest or continues moving at constant velocity according to Newton's first law. In the tip-to-tail view, the polyline returns to its starting point, forming a closed polygon. That idea, "stationary means zero resultant force," is foundational in structural design.
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Do you often use vector addition in CAE work?
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All the time. In finite element analysis, multiple forces often meet at a node and must be summed. Load cases such as gravity plus wind pressure or thermal stress plus mechanical loading are also vector combinations. In CFD, velocity and pressure vectors are combined to understand flow direction. Vector addition is one of the small ideas that quietly supports a lot of CAE interpretation.
Frequently Asked Questions
What is the difference between vector addition and vector decomposition?
Addition is the operation of combining multiple vectors into one (F1+F2+… → F), while decomposition is splitting one vector into multiple components (F → Fx, Fy). Both are based on component calculations and are inverse operations. In CAE, they are used for load setting (decomposition) and result evaluation (addition).
Where is the angle reference (0° direction)?
In this simulator, following the standard polar coordinate system in mathematics, the positive x-axis direction (to the right) is 0°, and angles are measured counterclockwise. So 90° = upward, 180° = leftward, 270° = downward. In engineering, a north-referenced system (positive y-axis as 0°, clockwise) is sometimes used, so convert according to the problem definition.
Can the resultant force be zero without equilibrium?
Yes. Even if the resultant force (translational sum) is zero, if the sum of moments (rotational forces) is not zero, the object will rotate. This is called the balance of "moment of force" or "torque." Complete static equilibrium (rigid body balance) requires both ΣF=0 and ΣM=0.
How do you do 3D vector addition?
It's an extension of 2D: just add the z-component. Sum as Fx=ΣFᵢcosαᵢ, Fy=ΣFᵢcosβᵢ, Fz=ΣFᵢcosγᵢ (α,β,γ are direction cosines with x,y,z axes). The magnitude of the resultant is √(Fx²+Fy²+Fz²). In CAE 3D analysis, 3D vectors are naturally handled, with 6 degrees of freedom (forces in x,y,z and moments about three axes) at each node.
Is vector addition used in electrical circuits?
Yes, in AC circuits. When voltage and current are represented as complex numbers (phasors), adding voltages with different phases is exactly vector addition. They are decomposed into R component (real part = cosφ direction) and X component (imaginary part = sinφ direction) and added. The impedance Z=R+jX vector diagram (impedance triangle) uses the same principle as this simulator's visualization.
How are "dot product" and "cross product" different from vector addition?
The dot product A·B=|A||B|cosθ returns a scalar value and represents the "product of parallel components" between two vectors. It is used in work W=F·d (force and displacement). The cross product A×B returns a vector with magnitude |A||B|sinθ (area of parallelogram) and direction perpendicular to both vectors. It is used in torque τ=r×F. Both are different operations from "addition (summation)" and appear frequently in CAE stress-strain tensor calculations.
What is 2D Vector Addition?
The 2D Vector Addition Simulator covers a fundamental topic in engineering and applied physics. This interactive simulator lets you explore the key behaviors and relationships by directly manipulating parameters and observing real-time results.
By combining numerical computation with visual feedback, the simulator bridges the gap between abstract theory and physical intuition — making it an effective learning tool for students and a rapid-verification tool for practicing engineers.
Physical Model & Key Equations
The simulator is based on standard component summation in a Cartesian coordinate system. Understanding these equations is key to interpreting the results correctly.
Each parameter in the equations corresponds to a slider in the control panel. Moving a slider changes the equation's solution in real time, helping you build a direct connection between mathematical expressions and physical behavior.
Real-World Applications
Engineering Design: Vector addition is applied across mechanical, structural, electrical, and fluid engineering disciplines. This tool provides a quick way to estimate design parameters and sensitivity before committing to full CAE analysis.
Education & Research: Widely used in engineering curricula to connect theory with numerical computation. Also serves as a first-pass validation tool in research settings.
CAE Workflow Integration: Before running finite element (FEM) or computational fluid dynamics (CFD) simulations, engineers use simplified models like this to establish physical scale, identify dominant parameters, and define realistic boundary conditions.
Common Misconceptions and Points of Caution
Model assumptions: The mathematical model used here relies on simplifying assumptions such as linearity, homogeneity, and isotropy. Always verify that your real system satisfies these assumptions before applying results directly to design decisions.
Units and scale: Many calculation errors arise from unit conversion mistakes or order-of-magnitude errors. Pay close attention to the units shown next to each parameter input.
Validating results: Always sanity-check simulator output against physical intuition or hand calculations. If a result seems unexpected, review your input parameters or verify with an independent method.