Calculate Vector Cross Products in 3D
Compute the cross product of two vectors in three-dimensional space. Our calculator provides the resultant vector perpendicular to both input vectors, essential for physics, engineering, and 3D graphics calculations.
Explore the fascinating world of cross products! This calculator helps you compute the cross product of two 3D vectors, resulting in a vector perpendicular to both inputs.
The cross product of two vectors a = (ax, ay, az) and b = (bx, by, bz) is defined as:
a × b = (ay*bz - az*by, az*bx - ax*bz, ax*by - ay*bx)
The resulting vector is perpendicular to both input vectors and its magnitude equals the area of the parallelogram spanned by the two vectors.
The cross product (vector product) of two vectors produces a third vector perpendicular to both original vectors. For vectors a = (a₁, a₂, a₃) and b = (b₁, b₂, b₃), the cross product a × b = (a₂b₃ - a₃b₂, a₃b₁ - a₁b₃, a₁b₂ - a₂b₁). The magnitude equals the area of the parallelogram formed by the two vectors.
The direction of the cross product follows the right-hand rule: point your index finger along the first vector, middle finger along the second vector, and your thumb points in the direction of the cross product. This determines the orientation of the resulting vector.
Cross products are crucial for: calculating torque (r × F), determining magnetic force on moving charges, finding angular momentum, computing surface normals in computer graphics, and analyzing rotational motion in mechanics.
