Vector Product or Cross Product

Subtopic 2 — Vector Product or Cross Product

Vector Product (Cross Product)

Physical Meaning of Cross Product

The cross product measures the perpendicular interaction between two vectors. Its result contains: A magnitude A direction The magnitude is: |A⃗ × B⃗| = AB sin θ The direction is determined using the right-hand rule.

Direction of Cross Product

The direction of A⃗ × B⃗ is perpendicular to both A⃗ and B⃗. Right-Hand Rule 1. Point the fingers of your right hand along A⃗. 2. Curl them toward B⃗ through the smaller angle. 3. Your thumb indicates the direction of A⃗ × B⃗. Changing the order reverses the direction: B⃗ × A⃗ = −(A⃗ × B⃗)

Cross Product at Special Angles

Parallel Vectors When θ = 0°: |A⃗ × B⃗| = AB sin 0° = 0 Perpendicular Vectors When θ = 90°: |A⃗ × B⃗| = AB sin 90° = AB The cross product has its maximum magnitude. Opposite Vectors When θ = 180°: |A⃗ × B⃗| = AB sin 180° = 0

Cross Products of Unit Vectors

The basic cyclic order is: î → ĵ → k̂ → î Following this order gives a positive result: î × ĵ = k̂ ĵ × k̂ = î k̂ × î = ĵ Reversing the order gives a negative result: ĵ × î = −k̂ k̂ × ĵ = −î î × k̂ = −ĵ The cross product of a unit vector with itself is zero: î × î = 0 ĵ × ĵ = 0 k̂ × k̂ = 0

Cross Product Using Components

Consider: A⃗ = Aₓ î + Aᵧ ĵ + Az k̂ B⃗ = Bₓ î + Bᵧ ĵ + Bz k̂ Then: A⃗ × B⃗ = (AᵧBz − AzBᵧ) î + (AzBₓ − AₓBz) ĵ + (AₓBᵧ − AᵧBₓ) k̂ It may also be written as the determinant with rows (î ĵ k̂), (Aₓ Aᵧ Az), (Bₓ Bᵧ Bz).

Properties of Cross Product

Anti-Commutative Property A⃗ × B⃗ = −(B⃗ × A⃗) Distributive Property A⃗ × (B⃗ + C⃗) = A⃗ × B⃗ + A⃗ × C⃗ Cross Product of a Vector With Itself A⃗ × A⃗ = 0 Scalar Multiplication (mA⃗) × (nB⃗) = mn(A⃗ × B⃗) Not Generally Associative Usually: A⃗ × (B⃗ × C⃗) ≠ (A⃗ × B⃗) × C⃗

Area Represented by Cross Product

The magnitude of a cross product gives the area of a parallelogram formed by two vectors: Area of parallelogram = |A⃗ × B⃗| The area of a triangle formed by the same vectors is: Area of triangle = (12)|A⃗ × B⃗|

Applications of Cross Product

Applications

• Torque: τ⃗ = r⃗ × F⃗ ; τ = rF sin θ
• Angular Momentum: L⃗ = r⃗ × p⃗
• Magnetic Force on a Charge: F⃗ = q(v⃗ × B⃗)
• Magnetic Force on a Wire: F⃗ = I(L⃗ × B⃗)
• Area of a Parallelogram: A = |a⃗ × b⃗|

Solved Examples — Cross Product