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Vector Addition & Subtraction

2D & 3D Vector Resultant Calculator

Calculate the resultant vector, magnitude, and unit vector components.

Vector A
Vector B

Resultant Vector

Calculation results will appear here.

Ultimate Guide to Vector Operations

What is a Vector?

In physics and mathematics, a vector is a geometric object that has both magnitude (size) and direction. Unlike scalar quantities (like temperature or mass) which only have size, vectors are used to describe quantities like force, velocity, acceleration, and displacement.

Component Form:

A = (Ax, Ay, Az)

Vector Addition Explained

To add two vectors, we use the head-to-tail method or simply add their corresponding components. If A = (Ax, Ay, Az) and B = (Bx, By, Bz), then the resultant R is:

  • Rx = Ax + Bx
  • Ry = Ay + By
  • Rz = Az + Bz

Vector Subtraction

Subtraction is just adding the negative of the second vector. Essentially, we subtract the components of Vector B from Vector A:

R = A - B = (Ax - Bx, Ay - By, Az - Bz)

Finding Magnitude

The magnitude (length) of a vector |A| is calculated using the distance formula (Pythagorean theorem) in 3D space:

|A| = √(Ax² + Ay² + Az²)

Frequently Asked Questions

A unit vector is a vector that has a magnitude of exactly 1. It is often created by dividing a vector by its own magnitude. It represents only the direction of the original vector.

Simply set the Z-component to 0. A 2D vector (x, y) is equivalent to a 3D vector (x, y, 0).

It allows physicists to calculate the net force acting on an object, the resulting velocity of a moving body influenced by multiple factors, or the total displacement after multiple movements.

No. Magnitude represents length or size, which is always a non-negative value (≥ 0).

The resultant vector is the sum or difference of two or more individual vectors. It represents the combined effect of all the component vectors.

Physics Tips

  • Use the head-to-tail rule for visualization.
  • Direction is relative to the origin (0,0,0).
  • For displacement, order of addition doesn't matter.
  • Watch out for negative coordinates!

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Disclaimer

Calculations are based on standard Euclidean geometry. Use for academic and general purposes. Double-check for mission-critical engineering.