## 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 = (a2*b3 - a3*b2, a3*b1 - a1*b3, a1*b2 - a2*b1)**

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 = (2*6 - 3*5, 3*4 - 1*6, 1*5 - 2*4) = **(-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)