Vector Calculator

Whether you’re balancing forces in a physics lab or solving a linear algebra problem, manual vector arithmetic can be slow and error-prone. The vector calculator on this page handles the heavy lifting: it instantly computes sums, differences, dot products, cross products, magnitudes, and angles for two- and three-dimensional vectors. All you need are the Cartesian components.

How Does a Vector Calculator Work?

The vector calculator uses Cartesian coordinates–the most familiar notation. For a 2D vector, you enter the horizontal (x) and vertical (y) components; for 3D, you add the depth (z) component. After entering two vectors A and B, you select the operation you need. The tool processes the components using standard vector algebra formulas and displays the result immediately, with no page reloads.

Inputs are parsed as real numbers, supporting integers and decimals. The calculator presents both the final numeric values and, where relevant, the component form of the answer. This makes it equally useful for quick checks and for learning the underlying math.

Core Vector Operations You Can Compute

Below are the main calculations the vector calculator performs. While 2D mode omits the z‑component and cross product, all other operations work in both dimensions.

• Addition: A + B = (Ax + Bx, Ay + By, Az + Bz)
  The result is the vector that completes the triangle when drawn from the tail of A to the head of B.

• Subtraction: A – B = (Ax – Bx, Ay – By, Az – Bz)
  Useful for finding displacement between two positions.

• Scalar multiplication: k A = (k·Ax, k·Ay, k·Az)
  Scaling a vector changes its length by factor k without altering its direction.

• Dot product: A·B = Ax·Bx + Ay·By + Az·Bz
  Produces a single number (scalar). It measures the projection of one vector onto another and is used to find the angle between them: cos θ = (A·B) / (|A|·|B|).

• Cross product (3D only): A×B = (Ay·Bz – Az·By, Az·Bx – Ax·Bz, Ax·By – Ay·Bx)
  Returns a vector perpendicular to both A and B, with magnitude equal to the area of the parallelogram they span.

• Magnitude: |A| = √(Ax² + Ay² + Az²)
  The length of the vector. In physics, this might represent speed or force intensity.

• Unit vector: \̂A = (Ax/|A|, Ay/|A|, Az/|A|)
  A vector with magnitude 1 that points in the same direction as A.

All these computations are performed by the calculator to machine precision, so you can focus on interpreting the results.

2D and 3D Vector Calculation Examples

2D example
Let A = (3, 4) and B = (1, 2).

• Addition: A + B = (4, 6)
• Magnitude of A: |A| = √(3² + 4²) = 5
• Dot product: A·B = 3·1 + 4·2 = 11
• Angle between them: cos θ = 11 / (5·√5) ≈ 0.9839, θ ≈ 10.3°

3D example
Let A = (1, 2, 3) and B = (4, 5, 6).

• Cross product: A×B = (2·6 – 3·5, 3·4 – 1·6, 1·5 – 2·4) = (–3, 6, –3)
• Magnitude of A×B: √(9 + 36 + 9) = √54 ≈ 7.35

These examples show how quickly the calculator reduces a page of manual arithmetic to a single step.

When to Use an Online Vector Tool

An online vector calculator saves time when:

• You are decomposing a force into components and need the resultant.
• You are computing the angle between two directions in navigation or 3D modeling.
• You want to verify homework problems in vectors and linear algebra.
• You work with cross products frequently and want to avoid sign errors.

Because the tool handles all operations in one place, you can compare results side by side and catch mistakes that are easy to miss when doing calculations by hand.

This calculator is provided for educational and reference purposes. For critical engineering or scientific applications, always verify results with manual calculations or dedicated software.