Final answer:
The unit tangent and unit normal vectors for the given vector function r(t) can be found by first taking the derivative of r(t) to get the velocity vector, normalizing it for the unit tangent vector, and then differentiating and normalizing the unit tangent vector for the unit normal vector.
Step-by-step explanation:
To find the unit tangent and unit normal vectors for the vector function r(t) = 6t, 1 + 2t², t², we first need to find the tangent vector, which is the derivative of r(t) with respect to t. The derivative r'(t) = dr(t)/dt gives the velocity vector v(t), which is v(t) = 6, 4t, 2t. To get the unit tangent vector t(t), we normalize v(t) by dividing it by its magnitude, |v(t)|.
After computing |v(t)|, we divide each component of v(t) by this magnitude to obtain t(t). To find the normal vector n(t), we need to take the derivative of the unit tangent vector t(t) and then normalize that derivative. This normalized derivative becomes the unit normal vector n(t).