Condition for constant direction of vector

A vector a(t) has constant direction when its unit vector â = a/|a| does not change with t. Assume a ≠ 0.

Write a = r â, where r = |a| is a scalar function of t and â is a unit vector.

Then da/dt = r′â + r (dâ/dt), where r′ = dr/dt, so

a × da/dt = r r′ (â × â) + r² (â × dâ/dt) = r² (â × dâ/dt). ... (1)

Necessary condition

If the direction is constant, dâ/dt = 0. Then from (1), a × da/dt = 0.

(Directly: a = f(t)â₀ with â₀ constant gives da/dt = f′â₀, parallel to a, so the cross product is zero.)

Sufficient condition

Let a × da/dt = 0. By (1), â × dâ/dt = 0 (as r ≠ 0).

Since â · â = 1, differentiating gives 2â · dâ/dt = 0, so dâ/dt is perpendicular to â. Therefore

|â × dâ/dt| = |â||dâ/dt| sin 90° = |dâ/dt|.

But â × dâ/dt = 0, so |dâ/dt| = 0, i.e. dâ/dt = 0. Hence â is constant: a has constant direction.

Therefore a has constant direction if and only if a × da/dt = 0.