3D Vector Cross Product Calculator

3D Vector Cross Product Calculator | High Precision 2026

Compute the orthogonal resultant vector of two 3D vectors (A & B) instantly. This tool utilizes the determinant method to calculate the cross product, magnitude, and direction, essential for physics, engineering, and 3D computer graphics. Optimized for 2026 standards.

Vector A

Ax:

Ay:

Az:

Vector B

Bx:

By:

Bz:

Please enter valid numeric values for all components.

Resulting Vector (A × B)

C = < 0, 0, 0 >

Magnitude |C|

Angle (θ)

0°

Understanding the 3D Vector Cross Product

In mathematics and physics, the cross product (or vector product) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space. Unlike the dot product, which results in a scalar, the cross product results in a vector that is perpendicular to both input vectors A and B. This makes it an indispensable tool for calculating torque, magnetic forces, and surface normals in 3D modeling.

The Mathematical Formula

The cross product of two vectors $ \mathbf{A} = (A_x, A_y, A_z) $ and $ \mathbf{B} = (B_x, B_y, B_z) $ is defined by the determinant of a matrix:

$\mathbf{C} = \mathbf{A} \times \mathbf{B} = (A_yB_z - A_zB_y, A_zB_x - A_xB_z, A_xB_y - A_yB_x)$

Our calculator implements this formula using 64-bit precision to ensure that even the smallest scientific measurements are handled accurately, following the IEEE 754-2019 standard.

Importance of Cross Product in Engineering

Engineers use the cross product to determine the moment of a force about a point, which is crucial for structural stability. In the field of robotics, it helps in calculating the velocity of various links in a robotic arm. In computer science, specifically game development and 3D rendering, the cross product is used to find the "normal" vector of a polygon. This normal vector determines how light bounces off a surface, creating realistic shadows and highlights.

How to Use This Calculator

Input Vector A: Enter the x, y, and z components into the red-labeled fields.
Input Vector B: Enter the corresponding components for the second vector in the blue fields.
Click Calculate: The engine will compute the coordinates of the new vector C.
Analyze Visuals: The magnitude bar shows the strength of the resulting vector relative to a standard unit, while the angle indicates the separation between inputs.

Key Properties of Cross Products

Anti-commutativity: $ \mathbf{A} \times \mathbf{B} = -(\mathbf{B} \times \mathbf{A}) $. Changing the order reverses the direction.
Parallel Vectors: If two vectors are parallel or collinear, their cross product is zero.
Right-Hand Rule: The direction of the resulting vector follows the right-hand rule—point your fingers from A to B, and your thumb points toward C.