Lesson 5: Cross Product

• Cross Product


  • It is used to calculate angles of vectors, areas of parallelograms, and vectors perpendicular to the vectors being multiplied. It has applications in mathematics, engineering, and physics
• Area of parallelogram using cross product

  The magnitude of the cross product of two vectors is equal to the area of the parallelogram, which is calculated as base x height
• Magnitude of the cross product of two vectors


  |a x b| = |a| |b| sin θ, where θ is the angle between the vectors, 0 ≤ θ ≤ 180° (when placed tail to tail)
• Unlike the dot product which gives us a scalar quantity, the cross product gives us a vector quantity
• The cross product of two vectors 𝑢 and 𝑣⃗ is a vector that is perpendicular to both vectors 𝑢 and 𝑣⃗
• The cross product can only be done in 3-space (ℝ3)
• The direction of 𝑢 × 𝑣⃗ is perpendicular to the plane containing 𝑢 and 𝑣⃗ such that 𝑢, 𝑣⃗ and 𝑢 × 𝑣⃗ form a right-handed system
• Method 1 to find the direction of the cross product

  Curl your right-hand fingers in the direction of rotation from 𝑢 to 𝑣⃗ (𝜃 < 180°). Your thumb points in the direction of 𝑢 × 𝑣⃗
• Method 2 to find the direction of the cross product

  Point your thumb to the first vector (𝑢) and the rest of the fingers to the 2nd vector (𝑣⃗), the direction of your palm will be the direction of the cross product (𝑢 × 𝑣⃗)
• Cross product formula

  𝑎⃗ × 𝑏 = |𝑎⃗||𝑏| sin 𝜃 n, where 𝜃 is the angle between the vectors, 0 ≤ 𝜃 ≤ 180° and 𝑛 is the unit vector orthogonal to both 𝑎⃗ and 𝑏 following the right-hand rule for direction
• Some things to think about: |a × b| = |b × a|
• Cross product of two vectors

  (a2b3 − a3b2, a3b1 − a1b3, a1b2 - a2b1)
• Cross product is NOT commutative
• 𝑎⃗ × 𝑏 = −(b × a)
• Anticommutative property

  𝑢 × 𝑣⃗ = −(𝑣⃗ × 𝑢)
• 𝑢 × 𝑣⃗ = −(𝑣⃗ × 𝑢) (anticommutative)
• 𝑢 × (𝑣⃗ + 𝑤) = 𝑢 × 𝑣⃗ + 𝑢 × 𝑤 (distributive)
• (𝑢 + 𝑣⃗) × 𝑤 = 𝑢 × 𝑤 + 𝑣⃗ × 𝑤 (distributive)
• 𝑘(𝑢 × 𝑣⃗) = (𝑘𝑢) × 𝑣⃗ = 𝑢 × (𝑘𝑣⃗) (associative)
• if 𝑢 and 𝑣⃗ are non-zero, 𝑢 × 𝑣⃗ = 0 iff 𝑢 and 𝑣⃗ are collinear
• For vectors 𝑢, 𝑣⃗ and 𝑤, the quantity (𝑢 × 𝑣⃗) ∙ 𝑤 is known as the triple scalar product
• The quantity (𝑢 × 𝑣⃗) × 𝑤 is known as the triple vector product