Final answer:
To find the vectors T and N and the binormal vector B for the given vector-valued function r(t) at a specific value of t, evaluate the given function at the given t value. The tangent vector T is the derivative of r(t) with respect to t, the normal vector N is the derivative of T with respect to t, and the binormal vector B is calculated by taking the cross product of T and N.
Step-by-step explanation:
To find the vectors T and N and the binormal vector B = T ⨯ N for the vector-valued function r(t), evaluate the given function at the given value of t.
Given r(t) = ti + t^2j + t^3/3k, and t0 = 1:
Step 1: Evaluate r(t) at t0:
r(1) = (1)(i) + (1^2)(j) + (1^3/3)(k) = i + j + (1/3)k.
Step 2: Tangent vector T is the derivative of r(t) with respect to t, evaluated at t0:
T(1) = r'(1) = (d/dt ti + d/dt(t^2)j + d/dt(t^3/3)k) = i + 2tj + t^2k evaluated at t = 1, so T(1) = i + 2j + k.
Step 3: Normal vector N is the derivative of T with respect to t, evaluated at t0:
N(1) = T'(1) = (d/dt i + d/dt(2t)j + d/dt(t^2)k) = 0i + 2j + 2tk evaluated at t = 1, so N(1) = 0i + 2j + 2k.
Step 4: Binormal vector B is calculated by taking the cross product of T and N:
B(1) = T(1) ⨯ N(1) = (1i + 2j + k) ⨯ (2j + 2k) = (-2j + 3k).