Cross Product Calculator

How to Calculate the Cross Product of Two Vectors

The cross product of two 3D vectors produces a third vector that is perpendicular to both input vectors. Unlike the dot product which returns a scalar, the cross product returns a vector whose magnitude equals the area of the parallelogram formed by the two original vectors. Our Cross Product Calculator computes this using the determinant formula with step-by-step expansion.

The Cross Product Formula

For two 3D vectors $\vec{a} = \langle a_1, a_2, a_3 \rangle$ and $\vec{b} = \langle b_1, b_2, b_3 \rangle$, the cross product is defined as the determinant of a 3x3 matrix with unit vectors $\hat{i}, \hat{j}, \hat{k}$ in the first row. Expanding by cofactors gives $\vec{a} \times \vec{b} = \hat{i}(a_2 b_3 - a_3 b_2) - \hat{j}(a_1 b_3 - a_3 b_1) + \hat{k}(a_1 b_2 - a_2 b_1)$.

Key Properties

The cross product is anti-commutative, meaning $\vec{a} \times \vec{b} = -(\vec{b} \times \vec{a})$. It is distributive over addition: $\vec{a} \times (\vec{b} + \vec{c}) = \vec{a} \times \vec{b} + \vec{a} \times \vec{c}$. When two vectors are parallel, their cross product is zero. The magnitude $|\vec{a} \times \vec{b}| = |\vec{a}||\vec{b}|\sin\theta$ equals the area of the parallelogram formed by the vectors.

Real-World Applications

The cross product appears throughout physics and engineering. Torque is calculated as $\tau = r \times F$, angular momentum as $L = r \times p$, and the Lorentz force on a charged particle in a magnetic field is $F = qv \times B$. In computer graphics, cross products are used to calculate surface normals for lighting calculations.

For related vector and matrix calculations, try the Dot Product Calculator, Vector Calculator, and Matrix Multiply Calculator.