Question

Find the vectors T, N, and B at the given point. r(t) = , 7 cos t, 7 sin t, 7 ln cos t. at the point (7,0,0). T= N= B=

Answer 1

To find the vectors T, N, and B at the given point, we can use the derivatives of the position vector and the cross product. The unit tangent vector T is (0, -7, 0), the unit normal vector N is (-1/√2, 0, -1/√2), and the unit binormal vector B is (0, 1/√2, 1/√2).

Step-by-step explanation:

To find the vectors T, N, and B at the given point, we need to find the unit tangent vector T, the unit normal vector N, and the unit binormal vector B.

Let's start with finding the unit tangent vector T. The unit tangent vector is the derivative of the position vector r(t) with respect to t, divided by its magnitude:

T = r'(t)/|r'(t)|

Next, we can find the unit normal vector N by taking the derivative of T with respect to t and dividing it by its magnitude:

N = T'/|T'|

Finally, the unit binormal vector B can be found by taking the cross product of T and N:

B = T x N

Substituting the values of r(t) and evaluating the derivatives at t = 0, we find:

T = (0, -7, 0)

N = (-1/√2, 0, -1/√2)

B = (0, 1/√2, 1/√2)