Vector Addition Calculator
Enter two 2D/3D vectors to compute their sum. a=(1,2,3), b=(4,5,6) → a+b=(5,7,9).
Input Data
Results
At a glance:Vector addition a+b follows the parallelogram law (or triangle law): placing the two vectors as adjacent sides of a parallelogram, the diagonal is the vector sum. Component-wise: a+b=(ax+bx, ay+by, az+bz). Properties: commutative a+b=b+a, associative (a+b)+c=a+(b+c), and a+0=a. Physically it expresses the superposition principle — composing forces (multiple forces acting together), velocities (object motion plus current/wind), and electric fields (multiple charges). |a+b|²=|a|²+|b|²+2a·b includes the dot-product term reflecting the angle between them.
Formula
Component addition: a+b = (ax+bx, ay+by, az+bz)
Parallelogram: |a+b|² = |a|²+|b|²+2|a||b|cosθ
Triangle law: place tail to head
Commutative: a+b = b+a
Zero vector: a+0 = a
$$\mathbf{a}+\mathbf{b}=(a_x+b_x,\;a_y+b_y,\;a_z+b_z), \quad |\mathbf{a}+\mathbf{b}|^2=|\mathbf{a}|^2+|\mathbf{b}|^2+2\mathbf{a}\cdot\mathbf{b}$$How to Use
- Enter the components ax, ay, az of vector a and bx, by, bz of vector b.
- The tool returns the component-wise sum (a+b) and its magnitude.
Case Studies
Force composition
Two forces a=(3,0) N and b=(0,4) N act on a body.
Resultant a+b=(3,4) N, magnitude 5 N at 53° to the x-axis.
This is the net force used in Newton's second law F=ma.
Velocity composition
A boat moves at a=(4,0) m/s across a river with current b=(0,3) m/s.
Ground velocity a+b=(4,3) m/s, speed 5 m/s.
Navigation adds wind/current vectors the same way.
FAQ
What is the parallelogram law?
Place two vectors as adjacent sides of a parallelogram; the diagonal from their common tail is the vector sum a+b. Equivalently, add corresponding components.
Is vector addition commutative?
Yes. a+b=b+a, both algebraically (component-wise) and geometrically (the same parallelogram). It is also associative.
How do I get the magnitude of a+b?
Compute the components first, then |a+b|=√((ax+bx)²+(ay+by)²+(az+bz)²). It also equals √(|a|²+|b|²+2|a||b|cosθ).
Where is it used?
Force composition, velocity composition (boat+current, plane+wind), electric/magnetic field superposition, structural analysis and navigation.
Related Tools
References
Content review: Calculatorism Science Team. Results are for reference only; please refer to the relevant authorities for the official figures.