Tangent line to parametrized curve examples

Example 1

Given the path (parametrized curve) c(t) = (3t + 2, t² -7, t - t²), find a parametrization of the line tangent to c(t) at the point c(1).

Solution: The line must pass through the point c(1) = (5, - 6, 0).

The derivative of the path is

When t = 1, the tangent vector must be

A parametrization for the tangent line is

l(t) = c(1) + (t - 1)c'(t₀) = (5i -6j) + (t - 1)(3i +2j - k) = (3t + 2)i + (2t - 8)j + (- t + 1)k

Example 2

Suppose a particle is following the path c(t) = (3t + 2, t² -7, t - t²). At time t₀ = 1, the particle flies off on a tangent. Compute the position of the particle at time t₁ = 2.

Solution: The key observation is that, after the time t₀, the position of the particle is given by tangent line l(t). From the solution to Example 1, we know that

The particle's position at time t₁ = 2 is