Cross Product Calculator

Unravel vector relationships with our interactive Cross Product Calculator. Visualize results and understand the math behind it.

Vector a

Vector b

Result:

Result will be displayed here after calculation.

Understanding Cross Product

The cross product, also known as the vector product, is a binary operation on two vectors in three-dimensional space. It results in a vector that is perpendicular to both of the input vectors.

Formula: For two vectors $\vec{a} = (a_x, a_y, a_z)$ and $\vec{b} = (b_x, b_y, b_z)$ , their cross product $\vec{a} \times \vec{b}$ is calculated as:

\vec{a} \times \vec{b} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ a_x & a_y & a_z \\ b_x & b_y & b_z \end{vmatrix} = (a_y b_z - a_z b_y)\mathbf{i} - (a_x b_z - a_z b_x)\mathbf{j} + (a_x b_y - a_y b_x)\mathbf{k}

Applications: The cross product has numerous applications in physics and engineering, including calculating torque, angular momentum, and magnetic forces. It is also used in computer graphics for surface normals and lighting calculations.

Learn more about cross product on Wikipedia.