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
Anticommutativity: a x b = -(b x a)
Distributivity: a x (b + c) = a x b + a x c
Scalar multiplication: (ca) x b = c(a x b)
Not associative: a x (b x c) != (a x b) x c in general
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)
## 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)