Final answer:
To find the unit tangent vector at a given point, we need to differentiate the position vector and normalize it. When t = 3, the unit tangent vector is (2/3)i - (1/27)j + (2/3)k.
Step-by-step explanation:
To find the unit tangent vector at the given point, we need to calculate the derivative of the position vector r(t) and normalize it. The position vector is r(t) = (t^2 - 2t)i + (1/3t)j + (1/3t^3 - 1/2t^2)k. Differentiating each component, we get r'(t) = 2ti - (1/3t^2)j + (t^2 - t)k. To normalize the tangent vector, we divide r'(t) by its magnitude. When t = 3, the unit tangent vector is t(3) = (2/3)i - (1/27)j + (2/3)k.