Vector Addition
Add vectors component-wise: A+B = (aₓ+bₓ, aᵧ+bᵧ, a_z+b_z). Parallelogram law: sum is the diagonal. Resultant magnitude |A+B| from Pythagorean theorem. Used in physics (forces) and graphics.
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Parallelogram law: A+B is diagonal. Commutative: A+B = B+A. Triangle inequality: |A+B| ≤ |A|+|B|.
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Why: Vector addition is fundamental in physics (force composition), computer graphics (position + velocity), and navigation (displacement). Parallelogram law gives geometric interpretation.
How: Add corresponding components. Magnitude of sum: √[(aₓ+bₓ)²+(aᵧ+bᵧ)²+(a_z+b_z)²]. Angle from dot product. Difference A−B = A + (−B).
Run the calculator when you are ready.
Enter Two Vectors
Vector A
Vector B
For educational and informational purposes only. Verify with a qualified professional.
🧮 Fascinating Math Facts
A+B = (aₓ+bₓ, aᵧ+bᵧ, a_z+b_z).
— Component Addition
Parallelogram law: sum = diagonal.
— Geometry
Key Takeaways
- • Vector addition adds corresponding components: .
- • The parallelogram law: A + B is the diagonal of the parallelogram formed by A and B.
- • Addition is commutative: A + B = B + A.
- • The resultant magnitude is .
- • For perpendicular vectors, (Pythagorean).
Did You Know?
Forces add as vectors. The net force is the vector sum of all individual forces.
Wind + aircraft velocity = ground velocity. Vector addition is essential for flight planning.
Object positions, velocities, and accelerations are combined using vector addition.
Structural loads and moments are summed using the parallelogram law.
|A + B| ≤ |A| + |B|. Equality holds when A and B are parallel.
A + (-A) = 0. Every vector has an additive inverse.
Understanding Vector Addition
To add two vectors, add their corresponding components:
Geometrically, place B with its tail at the head of A. The sum is the vector from the tail of A to the head of B (triangle rule).
Expert Tips
Component Form
Always add x to x, y to y, z to z. Never mix components.
Perpendicular Vectors
When A ⊥ B, |A+B|² = |A|² + |B|². Use for right triangles.
Opposite Vectors
A + (-A) = 0. Useful for cancellation in proofs.
Order Independence
A + B = B + A. The parallelogram shows both paths give the same diagonal.
Frequently Asked Questions
How do I add two vectors?
Add corresponding components: (a_x+b_x, a_y+b_y, a_z+b_z).
What is the parallelogram law?
Place vectors tail-to-tail. The sum is the diagonal from the common tail to the opposite corner.
Is vector addition commutative?
Yes. A + B = B + A for all vectors.
How do I find the magnitude of the sum?
Use |A+B| = √[(a_x+b_x)² + (a_y+b_y)² + (a_z+b_z)²].
When is |A+B| = |A| + |B|?
Only when A and B point in the same direction (parallel and same sense).
What about vector subtraction?
A - B = A + (-B). Subtract components: (a_x-b_x, a_y-b_y, a_z-b_z).
Can I add 2D and 3D vectors?
Treat 2D as 3D with z=0. (3,4) + (1,2) = (4,6) with z=0+0=0.
How to Use This Calculator
- Enter components of vectors A and B (use 0 for z in 2D).
- Click "Calculate" to get the sum, difference, resultant magnitude, and angle.
- Review the visualization showing the parallelogram.
- Copy results for homework or reports.
Disclaimer: This calculator uses standard floating-point arithmetic. Results are suitable for educational and professional use.
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