A curve C in three dimensions is given parametrically by (x(t), y(t), z(t)), where t is a real parameter, with a ≤ t ≤ b. Show that the equation of the tangent line at a point P on this curve where t = t₀ is given by

where x₀ = x(t₀), x´0 = x´(t₀), and so on. Hence find the equation of the tangent line to the circular helix

at t = 1/4π and show that the length of the helix between t = 0 and t = 1/2π is πa/√2.

XXoy- Yo = xo Yo z - Zo zo