Vector (cross) product

For non-zero, non-parallel a and b at angle θ, a × b = |a||b| sin θ n̂, where n̂ is the unit vector perpendicular to both, chosen so that a, b, n̂ form a right-handed system: turn the fingers of the right hand from a to b and the thumb points along n̂. If a or b is 0, or they are parallel, a × b = 0.

The result is a vector. For non-zero a and b, a × b = 0 exactly when a and b are parallel (collinear); in particular a × a = 0.

Order matters: b × a = −(a × b). For the axes, î × ĵ = k̂, ĵ × k̂ = î, k̂ × î = ĵ, while ĵ × î = −k̂ and î × î = 0.

The angle from the cross product: sin θ = |a × b|/(|a||b|). The cross product distributes over addition, a × (b + c) = a × b + a × c, and λ(a × b) = (λa) × b = a × (λb).

In components, a × b is the determinant with rows (î, ĵ, k̂), (a₁, a₂, a₃), (b₁, b₂, b₃): (a₂b₃ − a₃b₂)î − (a₁b₃ − a₃b₁)ĵ + (a₁b₂ − a₂b₁)k̂.

Worked: (2î + ĵ + 3k̂) × (3î + 5ĵ − 2k̂) = −17î + 13ĵ + 7k̂, whose length is √507.

Worked: a vector perpendicular to both a + b and a − b, for a = î + ĵ + k̂ and b = î + 2ĵ + 3k̂, is (a + b) × (a − b) = −2î + 4ĵ − 2k̂; its length is √24 = 2√6, so a unit vector is (−î + 2ĵ − k̂)/√6, and its negative is the other one.