Formula & Calculator
Cross Product of Vectors
Calculates the vector cross product of two 3D vectors, producing a new vector perpendicular to both original vectors.
Interpretation
a×b = (a₂b₃−a₃b₂, a₃b₁−a₁b₃, a₁b₂−a₂b₁). Vector product perpendicular to both. Used in torque, angular momentum, and 3D geometry.
Variables
| Symbol | Quantity | Unit |
|---|---|---|
| a, b | Vectors being multiplied | |
| a1, a2, a3 | Components of vector a | |
| b1, b2, b3 | Components of vector b |
What it means
The cross product (or vector product) of two 3D vectors a and b produces a vector that is perpendicular to both. Its magnitude is |a||b| sin θ, and its direction is given by the right‑hand rule. The components are computed as above. The cross product is essential in physics for torque (τ = r×F) and angular momentum (L = r×p). In geometry, it is used to find normals to planes, to compute areas of parallelograms, and in computer graphics for surface normals. The cross product is only defined in 3D (and 7D). Understanding the cross product is crucial for vector calculus, electromagnetism, and robotics.
Worked example
Cross Product – Two Examples
Real‑World| Parameter | Value |
|---|---|
| Vector a | (1,0,0) |
| Vector b | (0,1,0) |
| Parameter | Value |
|---|---|
| Vector a | (1,2,3) |
| Vector b | (4,5,6) |
Common mistakes
- Cross product: a × b = (a₂b₃ − a₃b₂, a₃b₁ − a₁b₃, a₁b₂ − a₂b₁) – a vector.
- Anti‑commutative: a × b = − (b × a).
- Result: A vector perpendicular to both a and b (right‑hand rule).
- Magnitude: |a × b| = |a||b| sinθ – area of parallelogram.
- Zero vector: If a and b are parallel (θ=0 or π), cross product is zero.
Applications
The cross product of two vectors, a × b = (a₂b₃ − a₃b₂, a₃b₁ − a₁b₃, a₁b₂ − a₂b₁), yields a vector perpendicular to both input vectors, with magnitude equal to the area of the parallelogram they span. This operation is essential in physics and engineering for calculating torque, angular momentum, and magnetic forces. In computer graphics, it determines surface normals for shading. In robotics, it defines rotational transformations. The cross product also appears in fluid dynamics (vorticity) and electromagnetism. By mastering the cross product, professionals can analyze 3D rotational effects and geometric properties that are critical in many engineering designs.
- Torque and moment calculations in mechanics
- Angular momentum and rotational dynamics
- Surface normal computation for rendering and collision detection
- Magnetic force on moving charges (Lorentz force)
- Vorticity and circulation in fluid dynamics
Frequently Asked Questions
The cross product of two 3D vectors a = (a₁, a₂, a₃) and b = (b₁, b₂, b₃) is a vector given by a × b = (a₂b₃ – a₃b₂, a₃b₁ – a₁b₃, a₁b₂ – a₂b₁). It is perpendicular to both a and b.
Applying the 3D cross product formula to 2D vectors without first treating them as 3D vectors with a zero third component.
|a × b| = |a| |b| sin θ, which is the area of the parallelogram formed by a and b. The direction is given by the right‑hand rule.
When the vectors are parallel (θ = 0 or π), so sin θ = 0.
a × a = 0.
- Anti‑commutative: a × b = –(b × a).
- Distributive: a × (b + c) = a × b + a × c.
- Scalar multiplication: (k a) × b = k (a × b).
- Finding a normal vector to a plane.
- Computing torque (τ = r × F).
- Calculating areas and volumes.
Area = |a × b|.
i × j = k, j × k = i, k × i = j.
a · (b × c) gives the volume of the parallelepiped formed by a, b, c.