Mathematics, Statistics & Geometry

Cross Product Calculator

Calculate the cross product of two 3D vectors instantly. Find the resulting perpendicular vector and its magnitude with a clear, step-by-step mathematical breakdown.

Cross Product Vector (C)
(-3, 6, -3)
C_x-3
C_y6
C_z-3
Magnitude ||C||7.348
Calculation StepsVector A = (1, 2, 3) Vector B = (4, 5, 6) Formula: C = A × B = (AyBz - AzBy, AzBx - AxBz, AxBy - AyBx) Cx = (2 * 6) - (3 * 5) = -3 Cy = (3 * 4) - (1 * 6) = 6 Cz = (1 * 5) - (2 * 4) = -3 Magnitude = √(Cx² + Cy² + Cz²) = √(-3² + 6² + -3²) = 7.348469

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3D Vector Math Made Easy

The Cross Product Calculator is an essential tool for physics, engineering, and 3D computer graphics. Whenever you need to find a direction that is strictly perpendicular to a given 2D surface or plane in 3D space, the cross product is the mathematical operation you need.

C=A×B=(AyBz−AzBy,AzBx−AxBz,AxBy−AyBx)\small \begin{aligned} \mathbf{C} = \mathbf{A} \times \mathbf{B} = (A_y B_z - A_z B_y, A_z B_x - A_x B_z, A_x B_y - A_y B_x) \end{aligned}

Where:
C\mathbf{C}=
The resulting orthogonal vector
A,B\mathbf{A}, \mathbf{B}=
The two 3D vectors being multiplied

Geometric Interpretation

The magnitude (length) of the cross product vector $|A \times B|$ is exactly equal to the area of the parallelogram formed by vectors A and B. This makes it incredibly useful for finding surface areas of 3D objects in computer rendering.

Practical Applications

  • Physics: Calculating torque (the cross product of the radius vector and the force vector).
  • Computer Graphics: Finding the "surface normal" of a 3D triangle to determine how light should bounce off it.
  • Electromagnetism: Calculating the magnetic force on a moving charge (Lorentz force).
  • Aerodynamics: Determining the axis of rotation for moving objects.

Frequently Asked Questions

The cross product of two vectors in 3D space is a third vector that is perpendicular (orthogonal) to both original vectors.

A dot product outputs a single scalar number representing how much two vectors point in the same direction. A cross product outputs a new vector representing the perpendicular direction.

If you point your index finger in the direction of vector A, and your middle finger in the direction of vector B, your thumb will point in the direction of the cross product vector C.

If two vectors are parallel or anti-parallel, their cross product is the zero vector (0,0,0) because there is no single perpendicular direction.

No. A × B = -(B × A). Reversing the order reverses the direction of the resulting vector.

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