Definition of cross product

Definition of Cross Product

The cross product (or vector product) is a binary operation on two vectors in three-dimensional space that produces a third vector perpendicular to both.

Definition

For vectors a = (a1, a2, a3) and b = (b1, b2, b3) in R^3, the cross product is:

a x b = (a2b3 - a3b2, a3b1 - a1b3, a1b2 - a2b1)

Equivalently, as a determinant:

a x b = det([[i, j, k], [a1, a2, a3], [b1, b2, b3]])

where i, j, k are the standard basis vectors.

Geometric Interpretation

• Direction: a x b is perpendicular to both a and b (right-hand rule)
• Magnitude: |a x b| = |a| * |b| * sin(theta) = area of the parallelogram spanned by a and b
• If a and b are parallel, then a x b = 0

Properties

1. Anticommutativity: a x b = -(b x a)
2. Distributivity: a x (b + c) = a x b + a x c
3. Scalar multiplication: (ca) x b = c(a x b)
4. Not associative: a x (b x c) != (a x b) x c in general
5. Triple product identity: a dot (b x c) = det([a, b, c]) (scalar triple product)

Example

For a = (1, 2, 3) and b = (4, 5, 6):

a x b = (26 - 35, 34 - 16, 15 - 24) = (-3, 6, -3)

Applications

• Physics: torque (tau = r x F), angular momentum (L = r x p)
• Computer graphics: computing surface normals
• Electromagnetism: Lorentz force (F = qv x B)