Vector product, torque and angular momentum

The vector (cross) product a × b is a vector:

• Its size is ab sin θ: zero for parallel vectors, greatest for perpendicular ones.
• Its direction is perpendicular to both a and b, given by the right-hand rule: point your fingers along a, curl them towards b, and your thumb shows a × b.
• Order matters: b × a = –a × b. And a × a = 0.

For the unit vectors, go round the circle î → ĵ → k̂ → î: î × ĵ = k̂, ĵ × k̂ = î, k̂ × î = ĵ. Going the other way round gives a minus sign (ĵ × î = –k̂).

In components, a × b is the determinant with î, ĵ, k̂ in the first row, a's components in the second and b's in the third. Example: a = 3î + 4ĵ and b = –3î + 7ĵ give a × b = (3 × 7 – 4 × (–3))k̂ = 33k̂. Both vectors lie in the x-y plane, so the answer points along z.

Two handy uses: |a × b| is the area of the parallelogram on a and b (33 square units here), so half of it is the area of the triangle; and a · (b × c) is the volume of the box on a, b, c.

Torque is the turning effect of a force about a point: τ = r × F, of size rF sin θ, measured in N m. Only the part of the force at right angles to r turns anything. Push a door 0.8 m from its hinges with 5 N at right angles, and τ = 4 N m; push at the hinge, or straight towards it, and nothing turns.

Angular momentum is the turning version of momentum: l = r × p. For a particle going round a circle of radius r at speed v, l = mvr: 0.5 kg at 4 m/s on a 2 m circle has l = 4 kg m² s⁻¹.

Just as F = dp/dt, τ = dl/dt. For a whole system, only external torques count: dL/dt = τ_ext.

So if the external torque is zero, the total angular momentum stays constant. Even a particle moving in a straight line at constant velocity has constant angular momentum about any point, because its distance r sin θ from the point, measured at right angles, never changes.