Three-Dimensional Vector Cross Product

The cross product (also called the vector product) is a fundamental binary operation on two vectors in three-dimensional Euclidean space. Unlike the dot product which yields a scalar, the cross product produces a new vector that is perpendicular to both original vectors and whose direction follows the right-hand rule.

Given two 3D vectors a = [a₁, a₂, a₃] and b = [b₁, b₂, b₃], their cross product a × b results in a vector c = [c₁, c₂, c₃] computed using the following formulas:

$$c_1 = a_2 \cdot b_3 - a_3 \cdot b_2$$ $$c_2 = a_3 \cdot b_1 - a_1 \cdot b_3$$ $$c_3 = a_1 \cdot b_2 - a_2 \cdot b_1$$

Key Properties of Cross Products:

• Perpendicularity: The resulting vector is orthogonal to both input vectors
• Anti-commutativity: (a \times b = -(b \times a)), meaning order matters
• Magnitude Interpretation: (|a \times b| = |a| \cdot |b| \cdot \sin(\theta)), where θ is the angle between vectors
• Area Calculation: The magnitude equals the area of the parallelogram formed by the two vectors
• Zero Result: Parallel or antiparallel vectors produce the zero vector [0, 0, 0]

This operation is ubiquitous in physics for computing torque and angular momentum, in computer graphics for determining surface normals, and in engineering for analyzing mechanical systems.

Your Task: Implement a function that takes two 3-dimensional vectors and returns their cross product as a new vector following the mathematical definition above.

a = [1, 0, 0]
b = [0, 1, 0]

[0, 0, 1]

a = [0, 1, 0]
b = [0, 0, 1]

[1, 0, 0]

a = [1, 2, 3]
b = [4, 5, 6]

[-3, 6, -3]