Question

Find the vectors t, n, and b at the given point for the function r(t) = 3 cos(t), 3 sin(t), 3 ln(cos(t)), at the point (3, 0, 0).

Answer 1

To find the vectors t, n, and b at the given point for the function r(t) = 3 cos(t), 3 sin(t), 3 ln(cos(t)), at the point (3, 0, 0), calculate the tangent vector, normal vector, and binormal vector by finding the derivative and cross product of the vector function.

Step-by-step explanation:

To find the vectors t, n, and b for the function r(t) = 3 cos(t), 3 sin(t), 3 ln(cos(t)) at the point (3, 0, 0), we need to calculate the tangent vector, normal vector, and binormal vector.

1. First, calculate the derivative of r(t) using the chain rule: r'(t) = (-3sin(t), 3cos(t), -3tan(t)sec(t)).
2. Next, evaluate r'(t) at t = 0 to find the tangent vector: r'(0) = (0, 3, 0).
3. Then, find the length of the tangent vector to normalize it: |r'(0)| = sqrt(0^2 + 3^2 + 0^2) = 3.
4. The normalized tangent vector is the unit tangent vector t: t = (0, 1, 0).
5. The normal vector n is found by taking the derivative of the unit tangent vector: n = (-d(t)/dt, d(t)/dt, -d(t)/dt) = (0, 0, 0).
6. The binormal vector b can be found by taking the cross product of t and n: b = t x n = (0, 0, 0).

Therefore, the vectors t, n, and b at the given point are t = (0, 1, 0), n = (0, 0, 0), and b = (0, 0, 0).