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

c'(t) = 3i + 2tj + (1 - 2t)k
When t = 1, the tangent vector must be
c'(1) = 3i + 2j - k.
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

l(t) = (3t + 2, 2t - 8,-t + 1).
The particle’s position at time t1 = 2 is
l(2) = (8,-4, 1).