Question

Consider the following vector function.

r(t) = (3t, 1/2t², t²)
(a) Find the unit tangent and unit normal vectors T(t) and N(t).

Answer 1

Step-by-step explanation:

r(t)=(3t, 12t2, t2),T(t),N(t)

i just know

Answer 2

To find the unit tangent T(t) and unit normal vectors N(t), calculate the normalized derivative of the position function for T(t), then differentiate T(t) and normalize it to get N(t).

Step-by-step explanation:

To find the unit tangent and unit normal vectors T(t) and N(t) for the given vector function r(t) = (3t, 1/2t², t²), we first need to find the derivative of r(t) with respect to t to get the velocity vector v(t). The unit tangent vector T(t) is the normalized velocity vector. Then, we calculate the derivative of T(t) with respect to t and normalize it to get the unit normal vector N(t).

1. First, we compute the velocity vector: v(t) = r'(t) = (3, t, 2t).
2. Then, we normalize v(t) to get the unit tangent vector: T(t) = v(t) / ||v(t)||.
3. After that, we find the derivative of T(t) to obtain T'(t).
4. Finally, we normalize T'(t) to get the unit normal vector: N(t) = T'(t) / ||T'(t)||.

Following these steps will provide the student with T(t) and N(t).