About Cross Product

The cross product (also called vector product) is a binary operation on two vectors in three-dimensional space. It results in a vector that is perpendicular to both input vectors.

Key Concepts:

• Cross Product: A × B = (a₂b₃ - a₃b₂, a₃b₁ - a₁b₃, a₁b₂ - a₂b₁)
• Magnitude: |A × B| = |A| |B| sin(θ)
• Direction: Perpendicular to both input vectors
• Right-hand Rule: Determines the direction of the result

Properties of Cross Product:

• Anti-commutative: A × B = -(B × A)
• Distributive: A × (B + C) = A × B + A × C
• Scalar multiplication: (kA) × B = k(A × B)
• Zero vector if vectors are parallel

Applications:

• Calculating torque in physics
• Finding normal vectors in computer graphics
• Determining angular momentum
• Solving geometric problems in 3D space

Tips:

• Input vectors must be 3D
• Result is perpendicular to both input vectors
• Use right-hand rule to determine direction
• Magnitude equals area of parallelogram formed by vectors