Vector Addition Calculator

Add two vectors A + B and find the resultant vector, its magnitude, and direction.

Enter Vector Components

Vector A
Vector B

Result

Resultant Vector A + B
--
magnitude: --
Vector A--
|A|--
Vector B--
|B|--
Resultant R = A + B--
|R|--
Direction of R--
Unit Vector of R--

Step-by-Step Solution

R = A + B = (Ax+Bx, Ay+By)

What is Vector Addition?

Vector addition is the operation of combining two or more vectors to produce a single resultant vector. The resultant represents the combined effect of both vectors. For example, if two forces act on an object, the resultant force is the vector sum of both individual forces.

Methods of Vector Addition

Component Method

Add corresponding components: Rx = Ax + Bx, Ry = Ay + By. This is the most precise method.

R = (Ax+Bx, Ay+By, Az+Bz)

Triangle Rule

Place the tail of vector B at the head of vector A. The resultant goes from the tail of A to the head of B.

Graphical: head-to-tail method

Parallelogram Rule

Place both vectors at a common point. The diagonal of the parallelogram they form is the resultant.

Graphical: common-tail method

Properties

Vector addition is commutative (A + B = B + A) and associative ((A + B) + C = A + (B + C)).

A + B = B + A

Practical Applications

  • Physics: Finding net force, combining velocities (e.g. boat in a current), resultant displacement.
  • Engineering: Structural load analysis, electrical circuit analysis (phasor addition).
  • Navigation: Combining wind velocity with aircraft velocity to find ground speed.
  • Computer Graphics: Translating objects, combining transformations, particle systems.

Vector Subtraction

Subtracting a vector is the same as adding its negative: A - B = A + (-B). To find -B, reverse the direction of B (negate all components).