What Is the Cross Product?
The cross product (also called the vector product) is a binary operation on two vectors in three-dimensional space that produces a third vector perpendicular to both input vectors. If A = (a₁, a₂, a₃) and B = (b₁, b₂, b₃), their cross product A × B = (a₂b₃ − a₃b₂, a₃b₁ − a₁b₃, a₁b₂ − a₂b₁). The magnitude of the resulting vector equals the area of the parallelogram formed by A and B, and its direction follows the right-hand rule.
Unlike the dot product which yields a scalar, the cross product yields a vector. This fundamental operation appears throughout physics and engineering: torque is the cross product of position and force, magnetic force is the cross product of velocity and magnetic field, and angular momentum is the cross product of position and momentum. The cross product is defined only for 3D vectors (and 7D vectors, though this is rarely used in practice).
The Cross Product Formula
Given vectors A = (a₁, a₂, a₃) and B = (b₁, b₂, b₃), the cross product is computed using the determinant of a 3×3 matrix with unit vectors i, j, k in the first row. The result is: A × B = (a₂b₃ − a₃b₂)i − (a₁b₃ − a₃b₁)j + (a₁b₂ − a₂b₁)k. Note the negative sign on the j component, which comes from the cofactor expansion of the determinant.
The magnitude of the cross product is |A × B| = |A| × |B| × sin(θ), where θ is the angle between the two vectors. This means parallel vectors have a zero cross product (sin 0° = 0), and perpendicular vectors have the maximum cross product magnitude (sin 90° = 1). This property makes the cross product useful for testing whether vectors are parallel.
Key Properties
Anticommutativity: A × B = −(B × A). Reversing the order reverses the direction of the resulting vector. This is unlike the dot product, which is commutative. Distributivity: A × (B + C) = A × B + A × C. The cross product distributes over addition. Scalar multiplication: (cA) × B = c(A × B). Scaling either vector scales the result proportionally.
Perpendicularity: The result A × B is always perpendicular to both A and B. This can be verified by computing the dot product: A · (A × B) = 0 and B · (A × B) = 0. Self cross product: A × A = 0 for any vector A, because the angle between a vector and itself is zero, making sin(θ) = 0.
Applications
Physics — Torque: Torque τ = r × F, where r is the position vector from the pivot point to the point of force application, and F is the force vector. The magnitude of torque determines rotational effectiveness, and its direction indicates the axis of rotation.
Physics — Electromagnetic force: The Lorentz force on a charged particle in a magnetic field is F = qv × B, where q is the charge, v is velocity, and B is the magnetic field. This cross product explains why charged particles move in circular or helical paths in magnetic fields.
Computer graphics: The cross product computes surface normals for lighting calculations. Given two edge vectors of a polygon, their cross product gives the normal vector, which determines how light reflects off the surface. This is fundamental to 3D rendering and shading algorithms.
Area calculation: The magnitude |A × B| equals the area of the parallelogram formed by vectors A and B. Half this value gives the area of the triangle formed by A and B. This provides an efficient method for computing areas of 3D polygons.
The Right-Hand Rule
The direction of A × B follows the right-hand rule: point your right hand's fingers in the direction of A, curl them toward B, and your thumb points in the direction of A × B. This convention ensures consistency across physics and engineering. If you reverse the order (B × A), the result points in the opposite direction, which is why the cross product is anticommutative.
Cross Product vs. Dot Product
The dot product A · B = a₁b₁ + a₂b₂ + a₃b₃ is a scalar that measures how much two vectors point in the same direction. It equals |A||B|cos(θ). The cross product A × B is a vector perpendicular to both with magnitude |A||B|sin(θ). Together, they provide complete information about the geometric relationship between two vectors: the dot product measures alignment, the cross product measures perpendicularity.