Vector Calculator
Instantly calculate vector magnitude, direction angle, dot product, cross product, and vector addition/subtraction in 2D and 3D space.
Calculate Vector Properties
Enter your vector components below to instantly compute magnitude, direction, and vector operations.
Vector Operations
Compute properties for 2D and 3D vectors.
How it Works
Understanding vector mathematics
Input Components
Enter the x, y, and (optionally) z components of your vectors. These represent the vector’s projection along each axis.
Select Operation
Choose whether you need basic properties (magnitude/direction) or operations involving two vectors (addition, dot, or cross product).
Instant Calculation
The calculator applies the appropriate algebraic or geometric formulas to compute the result instantly.
Review Details
View the primary result alongside supplementary data like individual magnitudes, unit vectors, and the angle between them.
Vector Formulas & Definitions
A quick overview of the mathematical formulas used in vector calculations.
| Operation | Formula | Result Type |
|---|---|---|
| Magnitude |A| | √(x² + y² + z²) | Scalar (Number) |
| Direction Angle (2D) | θ = atan2(y, x) | Scalar (Degrees/Radians) |
| Vector Addition | (Ax+Bx, Ay+By, Az+Bz) | Vector |
| Dot Product (A · B) | Ax·Bx + Ay·By + Az·Bz | Scalar (Number) |
| Cross Product (A × B) | (Ay·Bz – Az·By, Az·Bx – Ax·Bz, Ax·By – Ay·Bx) | Vector (Perpendicular to A & B) |
| Unit Vector (Â) | A / |A| | Vector (Magnitude of 1) |
Vector Calculator FAQ
Answers to the most frequently asked questions about vector mathematics and calculations.
A vector is a mathematical object that has both magnitude (size or length) and direction. It is commonly represented as an arrow in space, or algebraically as an ordered list of numbers (components), such as (x, y) in 2D or (x, y, z) in 3D.
The magnitude of a vector is calculated using the Pythagorean theorem. For a 2D vector (x, y), the magnitude is √(x² + y²). For a 3D vector (x, y, z), the magnitude is √(x² + y² + z²).
The dot product (scalar product) of two vectors results in a single number (scalar) and measures how much the vectors point in the same direction. The cross product (vector product) results in a new vector that is perpendicular to both original vectors, and its magnitude represents the area of the parallelogram they form.
No, the magnitude of a vector represents its length or size, which is always a non-negative value (zero or positive). However, the individual components (x, y, z) of a vector can be negative, which indicates direction along a specific axis.