Cross Product Definition

Trigonometry · Axiom Academy

A fundamental operation that creates a new vector perpendicular to two given vectors, with magnitude equal to the area of the parallelogram they form. Magnitude equals the area of the parallelogram formed by a and b Direction determined by the right-hand rule To calculate the cross product a × b , we use a determinant involving the standard basis vectors i , j , k : Where a = ⟨a₁, a₂, a₃⟩ and b = ⟨b₁, b₂, b₃⟩. This expands to give us each component of the result vector: The magnitude of the cross product has a beautiful geometric meaning: This also means: || a × b || = || a || · || b || · sin(θ), where θ is the angle between the vectors. The direction of a × b follows the right-hand rule :

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