Cross product

The cross product (or vector product) of a and b gives a vector:

a × b = |a| |b| sin θ n̂, where n̂ is a unit vector perpendicular to both a and b.

Its direction follows the right-hand rule: curl the fingers of your right hand from a towards b; your thumb points along a × b.

Working it out: write a 3 × 3 determinant with î, ĵ, k̂ in the top row, a's components in the second row, b's in the third, and expand along the top row.

Example: a = î + ĵ, b = ĵ + k̂. a × b = |î, ĵ, k̂; 1, 1, 0; 0, 1, 1| = î(1 – 0) – ĵ(1 – 0) + k̂(1 – 0) = î – ĵ + k̂. Check: it dots to 0 with both a and b.

Rules that differ from the dot product:

• Order matters: b × a = –(a × b).
• a × a = 0. More generally, for non-zero vectors, a × b = 0 means a and b are parallel (sin θ = 0).
• î × ĵ = k̂, ĵ × k̂ = î, k̂ × î = ĵ, going round the cycle; going backwards gives a minus sign.

Area: |a × b| = |a||b| sin θ is the area of the parallelogram with sides a and b. A triangle is half: ½|a × b|. In the example, |î – ĵ + k̂| = √3, so the parallelogram on a and b has area √3.

A unit vector perpendicular to both a and b is (a × b)/|a × b|. There are two answers: this one and its negative.