Final answer:
To find the principal unit normal vector to the curve r(t) = ti + 6tj at t=2, compute the normalized derivative of the curve to get the tangent, and then differentiate and normalize this tangent to obtain the principal unit normal vector.
Step-by-step explanation:
The principal unit normal vector of a curve at a given point provides insight into how the curve is turning at that point. In this case, finding the principal unit normal vector to the curve given by r(t) = ti + 6tj at t = 2 involves a few steps. First, we must find the tangent vector t(s) by differentiating the curve r(t) and normalizing it. Then, taking the derivative of that tangent vector will give us the change in the direction of the curve, from which we can get the normal vector n(t). This vector needs to be normalized to ensure it is a unit vector. The principal unit normal vector will be perpendicular to this curve's tangent at the specified parameter value of t = 2.