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Cross Product Calculator

Calculate the cross product of two three-dimensional vectors quickly and accurately. Enter the components of both vectors to get the resulting vector with a complete calculation breakdown.

Calculate Cross Product

Enter the X, Y, and Z components of two 3D vectors.

Vector A

Vector B

Cross Product Result

What Is a Cross Product?

The cross product is a mathematical operation performed on two three-dimensional vectors. Unlike the dot product, which produces a scalar value, the cross product produces a new vector.

The resulting vector is perpendicular to both of the original vectors. The direction of the resulting vector is determined using the right-hand rule, while its magnitude depends on the angle and magnitudes of the two original vectors.

A × B = (AᵧB𝓏 − A𝓏Bᵧ, A𝓏Bₓ − AₓB𝓏, AₓBᵧ − AᵧBₓ)

How Does the Cross Product Calculator Work?

This Cross Product Calculator uses the standard mathematical formula for calculating the vector cross product in three-dimensional space. Enter the X, Y, and Z values for both vectors, and the calculator automatically performs the required calculations.

  1. Enter the X, Y, and Z components of Vector A.
  2. Enter the corresponding X, Y, and Z components of Vector B.
  3. Calculate the first component using AᵧB𝓏 − A𝓏Bᵧ.
  4. Calculate the second component using A𝓏Bₓ − AₓB𝓏.
  5. Calculate the third component using AₓBᵧ − AᵧBₓ.
  6. Combine all three results to form the final cross product vector.

Cross Product Formula

Suppose the two vectors are:

A = (Aₓ, Aᵧ, A𝓏)
B = (Bₓ, Bᵧ, B𝓏)

The cross product is calculated using:

A × B =
(AᵧB𝓏 − A𝓏Bᵧ,
A𝓏Bₓ − AₓB𝓏,
AₓBᵧ − AᵧBₓ)

The final answer is always represented as a vector with three components.

Cross Product Calculation Example

Consider the following two vectors:

A = (1, 2, 3)
B = (4, 5, 6)

Using the cross product formula:

A × B =
(2×6 − 3×5,
3×4 − 1×6,
1×5 − 2×4)

After calculating each component:

A × B = (-3, 6, -3)

Therefore, the cross product of these two vectors is (-3, 6, -3).

Where Is the Cross Product Used?

Cross products are widely used in mathematics, physics, engineering, computer graphics, and three-dimensional geometry. They are useful whenever it is necessary to determine a vector perpendicular to two other vectors.

In physics, cross products can be used in calculations involving rotational motion, force, torque, and magnetic fields. In computer graphics, they are commonly used to calculate surface normals for lighting and 3D rendering.

Engineers and mathematicians also use cross products to solve geometric problems involving planes, directions, orientations, and three-dimensional coordinate systems.

Frequently Asked Questions

The cross product is a vector operation that produces a new vector perpendicular to both original three-dimensional vectors.
A dot product produces a scalar value, while a cross product produces a vector. The cross product is primarily used with three-dimensional vectors.
Yes. The cross product becomes the zero vector when two vectors are parallel or when one of the vectors is a zero vector.
Yes. The order matters because A × B is generally the opposite direction of B × A. Changing the order changes the sign of the resulting vector.
A zero cross product usually indicates that the two vectors are parallel, anti-parallel, or that one of the vectors has zero magnitude.
Yes. You can enter whole numbers, decimal numbers, and negative values for all vector components.