# Vector Calculator - Dot Product, Cross Product, Magnitude and Angle

> Free online vector calculator for 2D and 3D: addition, scalar multiples, dot and cross products, magnitude, angle, unit vector and projections, with checks.

- URL: https://qilingtools.com/en/math/vector
- Category: Math Tools (https://qilingtools.com/en/math)
- Tags: [Math](https://qilingtools.com/en/tags/math), [Physics](https://qilingtools.com/en/tags/physics)
- Last updated: 2026-10-26
- Price: Free, no sign-up
- Privacy: Runs locally in the browser; files are never uploaded

Works with 2D and 3D vectors: a+b, a−b, k·a, the dot and cross products, both magnitudes, the angle between them, the unit vector of a, the scalar and vector projection of a onto b, and whether the vectors are perpendicular or parallel. The cross product magnitude also gives the parallelogram and triangle areas.

## How to use

1. Choose 2D or 3D
2. Enter the components of a and b, and a scalar k if you need k·a
3. Read the results and copy any component

## Features

- Same tool for 2D and 3D; the 2D cross product is given as a scalar
- Dot product, cross product, magnitudes and the angle in degrees and radians
- Unit vector, scalar projection and vector projection
- Parallelogram and triangle areas from the cross product
- Automatic perpendicular / parallel / general classification

## FAQ

### What is the difference between the dot and cross products?

The dot product is a number, |a||b|cosθ, used for angles and projections. In 3D the cross product is a vector perpendicular to both, whose magnitude is the area of the parallelogram they span.

### How do I find the angle between two vectors?

cosθ = a·b / (|a||b|), then take the inverse cosine. Degrees and radians are both shown, and a zero vector is reported as undefined.

### How do I check whether vectors are perpendicular or parallel?

A dot product of 0 means perpendicular; a zero cross product (in 2D, x₁y₂−x₂y₁ = 0) means parallel. The tool states the conclusion for you.

### What is the scalar projection?

The length of a along b: a·b / |b|. Multiplying by the unit vector of b gives the vector projection, and both are computed.

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