Final answer:
To find the unit tangent and unit normal vectors, first find the derivative of the position function to get the velocity vector. Then normalize the velocity vector to get the unit tangent vector.
Step-by-step explanation:
To find the unit tangent vector, we first need to find the velocity vector. Taking the derivative of the position function gives us the velocity vector: v(t) = 9cos(t)î + 1ĵ - 9sin(t)ķ. Next, we find the magnitude of the velocity vector, which is |v(t)| = √(81cos²(t) + 1 + 81sin²(t)). The unit tangent vector is then t(t) = (1/|v(t)|) * v(t).
To find the unit normal vector, we need to take the derivative of the unit tangent vector with respect to t and then normalize it. The derivative of t(t) is dt/dt = -9sin(t)î + 0ĵ + 9cos(t)ķ. The magnitude of the derivative is |dt/dt| = √(81sin²(t) + 81cos²(t)). The unit normal vector is then n(t) = (1/|dt/dt|) * dt/dt.