MechanicsR = A + B

Vector Addition Simulator

Add two vectors graphically and analytically. Adjust magnitudes and angles to see the resultant vector update in real time.

— A (vector A)— B (vector B)- - R (resultant)

Vector A

Magnitude |A|

Angle θA

Vector B

Magnitude |B|

Angle θB

Resultant R

|R|0.000

θR0.0°

Rx0.000

Ry0.000

|A|5.0

θA30 °

|B|4.0

θB120 °

|R|0.000

θR0.0 °

About Vector Addition

A vector is a quantity that has both magnitude and direction — examples include displacement, velocity, force, and acceleration. Adding two vectors means combining their effects to find a single equivalent vector called the resultant.

ℹVectors can be added graphically using the tip-to-tail method, or analytically by summing their x and y components separately. Both methods give the same result.

Key Variables

Symbol	Name	Unit	Description
A	Magnitude of Vector A	dimensionless / m	Length of vector A
θA	Angle of Vector A	°	Angle from positive x-axis, counterclockwise positive
B	Magnitude of Vector B	dimensionless / m	Length of vector B
θB	Angle of Vector B	°	Angle from positive x-axis, counterclockwise positive
Ax, Ay	Components of A	same as A	Ax = A cosθA, Ay = A sinθA
Bx, By	Components of B	same as B	Bx = B cosθB, By = B sinθB
\|R\|	Magnitude of Resultant	same as A, B	\|R\| = √(Rx² + Ry²)
θR	Angle of Resultant	°	θR = atan2(Ry, Rx)

Component Method — Worked Example

Suppose A = 5 at θA = 30° and B = 3 at θB = 120°. Resolve into components:

A_{x} = 5 cos 30° = 4.33 A_{y} = 5 sin 30° = 2.50

B_{x} = 3 cos 120° = - 1.50 B_{y} = 3 sin 120° = 2.60

R_{x} = 4.33 + (- 1.50) = 2.83 R_{y} = 2.50 + 2.60 = 5.10

∣ R ∣ = 2.8 3^{2} + 5.1 0^{2} = 8.01 + 26.01 \approx 5.83

θ_{R} = arctan (\frac{5.10}{2.83}) \approx 61.0°

Graphical (Tip-to-Tail) Method

Draw vector A from the origin. Then draw vector B starting from the tip of A. The resultant R goes from the tail of A to the tip of B. The simulator draws A (blue) and B (green) head-to-tail with R (red) closing the triangle.

💡Try setting both vectors to the same angle — the resultant magnitude equals A + B. Try opposite angles — they partially or fully cancel.

Special Cases

Parallel vectors (same angle): |R| = A + B (maximum possible resultant).
Anti-parallel vectors (opposite angles): |R| = |A − B| (minimum possible resultant).
Perpendicular vectors: |R| = √(A² + B²), θR = atan(B/A).

Vector Addition Formulas

Component Decomposition

A_{x} = ∣ A ∣ cos θ_{A} A_{y} = ∣ A ∣ sin θ_{A}

Resultant Components

R_{x} = A_{x} + B_{x} R_{y} = A_{y} + B_{y}

Resultant Magnitude and Angle

∣ R ∣ = R_{x}^{2} + R_{y}^{2} θ_{R} = atan2 (R_{y}, R_{x})

Law of Cosines (Alternative)

∣ R ∣^{2} = A^{2} + B^{2} + 2 A B cos α

where α is the angle between the two vectors. This is equivalent to the component method.

Formula	Description	Notes
Rx = Ax + Bx	x-component of resultant	Sum of x-components
Ry = Ay + By	y-component of resultant	Sum of y-components
\|R\| = √(Rx²+Ry²)	Magnitude of resultant	Pythagorean theorem on components
θR = atan2(Ry, Rx)	Direction of resultant	Use atan2 to get correct quadrant
\|R\|max = A + B	Maximum resultant	When vectors are parallel
\|R\|min = \|A − B\|	Minimum resultant	When vectors are anti-parallel

ℹAlways use atan2(Ry, Rx) rather than atan(Ry/Rx) — atan2 correctly handles all four quadrants.

Frequently Asked Questions

Why do we add components separately?

x and y are perpendicular (orthogonal) directions, so they are independent. The x-component of the resultant is only affected by x-components of the added vectors, and similarly for y. This independence is the foundation of the component method.

What is the maximum possible resultant of two vectors?

The maximum resultant is A + B, achieved when both vectors point in the same direction (parallel). The minimum is |A − B|, achieved when they point in opposite directions.

Does the order of addition matter?

No. Vector addition is commutative: A + B = B + A. The resultant is the same regardless of which vector you draw first.

What is the angle convention used in this simulator?

Angles are measured counterclockwise from the positive x-axis (East). So 0° = East, 90° = North, 180° = West, 270° = South.

Can I add more than two vectors?

Yes, using the same component method. Add more x-components and more y-components together, then find the resultant magnitude and direction. The simulator shows two vectors for clarity.