Final answer:
To find T(t), N(t), and B(t) of the given curve, calculate the derivatives of r(t), normalize r'(t) to find T(t), find the derivative of T(t) to get N(t), and calculate B(t) using the cross product of T(t) and N(t).
Step-by-step explanation:
To find the unit tangent vector (T(t)), normal vector (N(t)), and binormal vector (B(t)) of the given curve, we need to calculate the derivatives of r(t).
1. First, find the derivative of r(t):
r'(t) = (2cos(t), 4, -2sin(t))
2. Normalize r'(t) to find T(t):
|T(t)| = ||r'(t)|| = sqrt((2cos(t))² + 4² + (-2sin(t))²) = sqrt(4cos²(t) + 16 + 4sin²(t)) = sqrt(20) = 2sqrt(5)
T(t) = (2cos(t)/(2sqrt(5)), 4/(2sqrt(5)), -2sin(t)/(2sqrt(5))) = (cos(t)/sqrt(5), 2/sqrt(5), -sin(t)/sqrt(5))
3. Find the derivative of T(t) to get N(t):
N(t) = T'(t) = (-sin(t)/sqrt(5), 0, -cos(t)/sqrt(5))
4. Calculate B(t) using the cross product of T(t) and N(t):
B(t) = T(t) × N(t) = (2/sqrt(5), -sin(t)/sqrt(5), cos(t)/sqrt(5))