Cross Products - Calculus II Project Home Page

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Cross Products
This page will help you understand the concept of a cross product and how to calculate one.

What is a cross product?
The cross product is an operation between two vectors that returns a vector orthogonal to the two vectors. (In case you forgot orthogonal means that u v = 0). Formally, the cross product takes two input vectors u = u₁i + u₂j + u₃k and v = v₁i + v₂j + v₃k.
It is defined to equal:
u × v = (u₂v₃ - u₃v₂)i - (u₁v₃ - u₃v₁)j + (u₁v₂ - u₂v₁)k

What's special about a cross product?
A cross product has six properties that can be useful in different situations.

u × v = -(v × u)
u × (v + w) = (u × v) + (u × w)
c(u × v) = (cu) × v = u × (cv)
u × 0 = 0 × u = 0
u × u = 0
u (v × w) = (u × v) w

Note: These properties assume c to be a scalar.

Where to from here?