Cross Product Calculator
Calculate the cross product of two 3D vectors and its magnitude.
Enter the components of two 3D vectors A and B to compute the cross product C = A × B, a vector perpendicular to both.
Vector A
Vector B
C = A × B =
Magnitude |C| =
C is perpendicular to both A and B.
The Cross Product Formula
The cross product of two 3D vectors A = (a₁, a₂, a₃) and B = (b₁, b₂, b₃) is a vector C = A × B given by the determinant form:
- C_x = a₂·b₃ − a₃·b₂
- C_y = a₃·b₁ − a₁·b₃
- C_z = a₁·b₂ − a₂·b₁
Equivalently, A × B is the symbolic determinant of the matrix whose rows are the unit vectors (i, j, k), the components of A, and the components of B.
The magnitude is |C| = √(C_x² + C_y² + C_z²) = |A|·|B|·sin θ, where θ is the angle between A and B.
Geometric Meaning
The cross product produces a vector that is perpendicular (orthogonal) to both input vectors. This makes it useful for finding the normal to a plane defined by two vectors.
The magnitude of the cross product equals the area of the parallelogram spanned by A and B. When A and B are parallel, the cross product is the zero vector, since the parallelogram collapses and sin θ = 0.
The Right-Hand Rule
The direction of A × B is determined by the right-hand rule: point the fingers of your right hand along A, curl them toward B, and your thumb points in the direction of the resulting vector C.
Note: The cross product is anti-commutative, meaning A × B = −(B × A). Swapping the order of the vectors reverses the direction of the result.
Related Calculators
Note: This calculator computes the cross product of two 3D vectors based on the formulas described. While we strive for accuracy, please verify important calculations independently. This tool is for educational and informational purposes and should not be the sole basis for academic or professional decisions.