Compute the cross product of two 3D vectors, with magnitude and direction information.
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Vector 1 (u)
Vector 2 (v)
What is the cross product?
The cross product of two 3D vectors produces a third vector that is perpendicular to both inputs, with a magnitude equal to the area of the parallelogram they span.
Right-hand rule
The direction of the resulting vector follows the right-hand rule: point your fingers along the first vector, curl them toward the second, and your thumb points in the direction of the cross product.
Result
The result vector (-3, 6, -3) is perpendicular to both input vectors.
Step-by-step
Also in Linear Algebra
Linear Algebra
A cross product calculator computes the vector perpendicular to two input vectors in three-dimensional space. The result gives both the direction and magnitude of the perpendicular vector.
For vectors u = (u₁, u₂, u₃) and v = (v₁, v₂, v₃), the cross product u × v = (u₂v₃ − u₃v₂, u₃v₁ − u₁v₃, u₁v₂ − u₂v₁).
The magnitude |u × v| equals |u||v|sin(θ), representing the area of the parallelogram formed by the two vectors.
u × v = (u₂v₃ − u₃v₂, u₃v₁ − u₁v₃, u₁v₂ − u₂v₁)
Cross product of 3D vectors.
Frequently asked questions
No — u × v = −(v × u). Swapping the operands reverses the direction of the result.
The cross product is only defined for 3D vectors. In 2D, a related scalar (u₁v₂ − u₂v₁) gives the signed area.
Related
These related calculators come from the same leaf category, nearby sibling categories, or the same top-level topic.
Calculate the dot product of two vectors in 2D or 3D, with angle and orthogonality detection.
Calculate the angle between two vectors in 2D or 3D using the dot product formula.
Calculate the magnitude of a 2D or 3D vector with squared magnitude, unit vector, and direction angles.
Normalize a 2D or 3D vector to find its unit vector (magnitude 1) with step-by-step work.
