Cross Product - Vector Product - Directed Area Product
- is a binary operation on two vectors in three-dimensional space
- is denoted by the symbol ×
- Given two linearly independent vectors 𝑣 and 𝑢, the cross product, 𝑣 × 𝑢, is a vector that is perpendicular to both 𝑣 and 𝑢 and thus normal to the plane containing them, and its magnitude is equal to the parallelogram produced by 𝑣 and 𝑢
- returns a pseudovector or more appropriately a 3D bivector
- only works in 3D and 7D
- similar to wedge product in which it can work in any dimension
Cross Product - Defined Computationally
Given two 3D vectors 𝑣 and 𝑢:
The cross product of 𝑣⨯𝑢 is defined as:
Cross Product - Defined Geometrically
What vector 𝑝=[𝑝1, 𝑝2, 𝑝3] has the property that:
(area of parallelogram defined by 𝑣 and 𝑢)(component of [𝑥, 𝑦, 𝑧] perpendicular to 𝑣 and 𝑢)
(area of parallelogram defined by 𝑣 and 𝑢)(magnitude of projection onto line perpendicular to 𝑣 and 𝑢)
same as
(area of parallelogram defined by 𝑣 and 𝑢)(taking dot-product of [𝑥, 𝑦, 𝑧] and a vector thats perpendicular to 𝑣 and 𝑢)
Cross Product - Properties
Given unit vectors 𝑖, 𝑗, 𝑘; the following holds:
- 𝑖 × 𝑗 = 𝑘
- 𝑗 × 𝑖 = -𝑘
- 𝑗 × 𝑘 = 𝑖
- 𝑘 × 𝑗 = -𝑖
- 𝑘 × 𝑖 = 𝑗
- 𝑖 × 𝑘 = -𝑗
non linear independence
- 𝑖 × 𝑖 = 0
- 𝑗 × 𝑗 = 0
- 𝑘 × 𝑘 = 0
𝑣 × 𝑢 = 𝑝, where:
- 𝑝‘s magnitude equals area of parallelogram of sides 𝑣 and 𝑢
- 𝑝 is perpendicular to 𝑣 and 𝑢