Cross Product Calculator

This calculator is designed to help students and professionals in linear algebra compute the cross product of two vectors quickly and accurately.

Calculator

Vector A

Vector B

Results

Cross Product: [ , , ]

Data Source and Methodology

All calculations are based on standard vector mathematics rules and linear algebra principles. Consult authoritative texts for more details.

The Formula Explained

\( \mathbf{A} \times \mathbf{B} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ A_x & A_y & A_z \\ B_x & B_y & B_z \end{vmatrix} \)

Glossary of Terms

Vector A: The first vector in the calculation.
Vector B: The second vector in the calculation.
Cross Product: The vector perpendicular to both A and B.

How It Works: A Step-by-Step Example

For example, if \( \mathbf{A} = [1, 2, 3] \) and \( \mathbf{B} = [4, 5, 6] \), the cross product is computed by evaluating the determinant as shown in the formula above.

Frequently Asked Questions (FAQ)

What is a cross product?

The cross product is a binary operation on two vectors in three-dimensional space, resulting in a vector that is perpendicular to both.

How is the cross product used?

It's used in physics and engineering to find a vector that is orthogonal to a plane defined by two vectors.

What are the properties of the cross product?

The cross product is anti-commutative, meaning \( \mathbf{A} \times \mathbf{B} = - (\mathbf{B} \times \mathbf{A}) \).

Can the cross product be calculated in 2D?

No, the cross product is specifically defined for 3D vectors.

What happens if the vectors are parallel?

If vectors are parallel, the cross product is a zero vector.

Tool developed by Ugo Candido. Content reviewed by experts in linear algebra. Last reviewed for accuracy on: October 2023.