Final answer:
To find the unit tangent vector T at t = π/6, differentiate each component of the helix function r(t) = (cos(-3t), sin(-3t), -4t) with respect to t, then substitute t = π/6 into the derivatives. The resulting unit tangent vector is T = (-3, 0, -4).
Step-by-step explanation:
To find the unit tangent vector T at t = π/6, we need to find the derivative of the helix function r(t) = (cos(-3t), sin(-3t), -4t). Let's differentiate each component of the function with respect to t first:
- x'(t) = 3sin(-3t)
- y'(t) = -3cos(-3t)
- z'(t) = -4
Now, substitute t = π/6 into the derivatives:
- x'(π/6) = 3sin(-3π/6) = 3sin(-π/2) = -3
- y'(π/6) = -3cos(-3π/6) = -3cos(-π/2) = 0
- z'(π/6) = -4
Therefore, the unit tangent vector T at t = π/6 is T = (-3, 0, -4).