3D Vector Cross Product Calculator
Calculate the cross product of two 3D vectors, returning a new vector perpendicular to both along with its magnitude. Used in physics, engineering, and 3D graphics.
About this calculator
The cross product U × V of vectors U = (u₁, u₂, u₃) and V = (v₁, v₂, v₃) produces a vector perpendicular to both. Its components are: (U×V)ₓ = u₂v₃ − u₃v₂, (U×V)_y = u₃v₁ − u₁v₃, (U×V)_z = u₁v₂ − u₂v₁. The magnitude is |U×V| = |U||V|sin(θ), where θ is the angle between the vectors. This equals the area of the parallelogram spanned by U and V. The direction follows the right-hand rule: curl the fingers of your right hand from U toward V and your thumb points in the direction of U×V. Unlike the dot product, the cross product is only defined in three (and seven) dimensions and produces a vector, not a scalar.
How to use
Let U = (1, 2, 3) and V = (4, 5, 6). Enter u₁=1, u₂=2, u₃=3, v₁=4, v₂=5, v₃=6. Compute each component: x = u₂v₃ − u₃v₂ = 2·6 − 3·5 = 12 − 15 = −3. y = u₃v₁ − u₁v₃ = 3·4 − 1·6 = 12 − 6 = 6. z = u₁v₂ − u₂v₁ = 1·5 − 2·4 = 5 − 8 = −3. So U×V = (−3, 6, −3). Magnitude = √(9 + 36 + 9) = √54 ≈ 7.35 units.
Frequently asked questions
What is the cross product used for in physics and engineering?
In physics, the cross product computes torque: τ = r × F, where r is the position vector and F is the applied force. It also gives angular momentum L = r × p, and the magnetic force on a charge F = q(v × B). In engineering, it is used to find surface normals for stress analysis and fluid dynamics. In 3D computer graphics, cross products compute face normals for lighting calculations and collision detection. Any time you need a vector orthogonal to two known vectors, the cross product is the standard tool.
How does the cross product differ from the dot product?
The dot product U·V = |U||V|cos(θ) produces a scalar and measures how much two vectors point in the same direction; it is zero when they are perpendicular. The cross product U×V produces a vector perpendicular to both inputs and has magnitude |U||V|sin(θ); it is zero when the vectors are parallel. The dot product is commutative (U·V = V·U), while the cross product is anti-commutative (U×V = −V×U). Use the dot product for projections and angles; use the cross product for areas, normals, and rotational quantities.
What does it mean when the cross product of two vectors is zero?
A zero cross product means the two vectors are parallel (or one of them is the zero vector). Since |U×V| = |U||V|sin(θ), the product is zero whenever sin(θ) = 0, i.e., θ = 0° or 180°. Geometrically, parallel vectors span no area — the parallelogram they form is completely flat. In practice, this is a quick test for collinearity: if three points A, B, C satisfy (B−A) × (C−A) = 0, the points are collinear. It also means the two vectors are linearly dependent, so they cannot form a basis for the plane.