Vector Cross Product Calculator

What is the vector cross product?

The vector cross product (also called the vector product) takes two three-dimensional vectors and returns a new vector that is perpendicular to both. The magnitude of the result equals the area of the parallelogram spanned by the two original vectors, and its direction follows the right-hand rule. This makes the cross product fundamental in physics, engineering, and 3D graphics for describing rotation, torque, and orientation in space.

If you have two vectors A and B in 3D, written in component form as

then the cross product is another vector, written as .

Cross product formulas

There are two common ways to express the cross product: a determinant form and a component-by-component formula. Both are mathematically equivalent.

Determinant form

A compact way to remember the cross product is as a 3 × 3 determinant involving the unit vectors i, j, k:

Component formula

Expanding that determinant leads to a straightforward component-wise formula. The result is another 3D vector whose components are

where

The calculator on this page implements exactly these formulas in JavaScript.

How to interpret the result

• Direction: The output vector is perpendicular to both input vectors. Use the right-hand rule: point your index finger along A, your middle finger along B, and your thumb indicates the direction of A × B.
• Magnitude: The size of the cross product equals |A||B|sin(θ), where θ is the angle between the vectors. This is equal to the area of the parallelogram spanned by A and B.
• Zero result: If the cross product is the zero vector, then the original vectors are parallel (or one of them is the zero vector).

Worked example

Suppose you have

• A = (1, 2, 3)
• B = (4, 5, 6)

Using the component formula:

• x-component: cy = ay · bz − az · by = 2 × 6 − 3 × 5 = 12 − 15 = -3
• y-component: cy = az · bx − ax · bz = 3 × 4 − 1 × 6 = 12 − 6 = 6
• z-component: cz = ax · by − ay · bx = 1 × 5 − 2 × 4 = 5 − 8 = -3

So

A × B = (-3, 6, -3).

If you enter these components into the calculator (A = 1, 2, 3 and B = 4, 5, 6), it will display exactly this result.

Cross product vs. dot product

The cross product is often compared with the dot product because both combine two vectors into a single quantity, but they behave very differently. The table below summarizes the key differences.

Common applications

• Torque: In mechanics, torque is defined as the cross product of the position vector from the pivot to the point of application and the force vector. The magnitude describes how strongly the force tends to cause rotation.
• Magnetic (Lorentz) force: In electromagnetism, the force on a charged particle moving in a magnetic field involves the cross product of velocity and magnetic field vectors.
• Surface normals in 3D graphics: Graphics engines use cross products to compute normal vectors to surfaces, which are then used for lighting and shading calculations.
• Area and orientation: The magnitude of the cross product gives the area of a parallelogram defined by two vectors, while its direction indicates the orientation in 3D space.

Assumptions and limitations of this calculator

• 3D vectors only: The tool is designed specifically for three-dimensional vectors with x, y, and z components. It does not handle 2D vectors or higher-dimensional vectors directly.
• Numeric inputs: Enter real-valued numbers (integers or decimals, positive or negative). Symbolic variables and expressions are not supported.
• Floating-point arithmetic: Calculations are performed in your browser using standard floating-point arithmetic. Extremely large or tiny values may be affected by rounding errors.
• Ordering matters: Note that A × B = −(B × A). Swapping the input vectors will flip the direction of the result.
• Privacy and offline use: All computations run locally in your browser; input values are not sent to a server. Once loaded, the calculator continues to work even without an active internet connection.

Using the calculator effectively

• Enter the x, y, and z components of Vector A and Vector B into the corresponding fields.
• Use negative signs where needed, and decimals for non-integer values.
• Click the calculate button to see the resulting vector components.
• If you get a zero vector, check whether your inputs describe parallel vectors or if a component was entered incorrectly.

This combination of formulas, geometric interpretation, and practical notes should help you both compute cross products correctly and understand what the result means in real-world contexts.