Answer:
the unit tangent vector T at t=4 is T₀ = ( 1/144 , 1/180 , 1)
Step-by-step explanation:
since r(t) = t² − 3*t, 1 + 4*t, 13*t³+ 12*t²
then the tangent vector will be r'(t) =dr/dt
r'(t) =dr/dt = 2*t-3 , 4 , 39*t² + 24*t
denoting r₀ , r'₀ as the vectors at t=4
r'₀=r'(4)= (2*4-3 , 4 , 39*4² + 24*4) = (5,4, 720)
the modulus of the vector r'₀ will be
|r'₀| = √(5²+4²+720²) = 720.028 ≈ 720
then the tangent vector will be
T₀ = r'₀/|r'₀| =(1/720) * (5,4, 720) = ( 1/144 , 1/180 , 1)
T₀ = ( 1/144 , 1/180 , 1)
