Question

Find the vectors T,N, and B at the given point.

r(t)= (8 cos t, 8 sin t, 8 ln cos t), (8,0,0)
T= ___
N= ___
B= ___

Answer 1

To find the vectors T, N, and B at the given point r(t) = (8 cos t, 8 sin t, 8 ln cos t), (8,0,0), we need to calculate the derivative of the position vector r(t) with respect to time, which will give us the velocity vector V(t).

Step-by-step explanation:

To find the vectors T, N, and B at the given point r(t) = (8 cos t, 8 sin t, 8 ln cos t), (8,0,0), we need to calculate the derivative of the position vector r(t) with respect to time, which will give us the velocity vector V(t).

V(t) = (dx/dt, dy/dt, dz/dt)

Using the chain rule, we can find the derivatives:

dx/dt = -8 sin t, dy/dt = 8 cos t, dz/dt = -8 tan t sec t

Therefore, the vector T is the unit vector in the direction of V(t), the vector B is the unit vector in the direction of the derivative of T, and the vector N is the unit vector that is orthogonal to both T and B.