I prefer to use brackets for the vectors rather than having all the i, j, k letters floating around. I'm also using exaggerated spacing in the notation for r ' (t) and r ' ' (t) so things can be seen clearly. Steps of finding the cross products are omitted because this was getting too long to type out.

a) r(t) = ⟨ 6t^2 , 6t - 2t^3 ⟩

r ' (t) = ⟨ 12t , 6 - 6t^2 ⟩

r ' ' (t) = ⟨ 12 , -12t ⟩

| r ' (t) | = √[(12t)^2 + (6 - 6t^2)^2]

= √(144t^2 + 36 - 72t^2 + 36t^4)

= √(36t^4 + 72t^2 + 36)

= √[(6t^2 + 6)^2]

= (6t^2 + 6)

aT = [ r ' (t) • r ' ' (t) ] / | r ' (t) |

= [ ⟨ 12t , 6 - 6t^2 ⟩ • ⟨ 12 , -12t ⟩ ] / (6t^2 + 6)

= (144t - 72t + 72t^3) / (6t^2 + 6)

= (72t^3 + 72t) / (6t^2 + 6)

= [12t * (6t^2 + 6)] / (6t^2 + 6)

= 12t

---

aN = | r ' (t) x r ' ' (t) | / | r ' (t) |

= (-72t^2 - 72) / (6t^2 + 6)

= [-12(6t^2 + 6)] / (6t^2 + 6)

= -12

-----------

b) r(t) = ⟨ 3t , 3t^2 , (1/2)t^2 ⟩

r ' (t) = ⟨ 3 , 6t , t ⟩

r ' ' (t) = ⟨ 0 , 6 , 1 ⟩

| r ' (t) | = √[(3)^2 + (6t)^2 + (t)^2] = √(37t^2 + 9)

aT = [ r ' (t) • r ' ' (t) ] / | r ' (t) |

= [⟨ 3 , 6t , t ⟩ • ⟨ 0 , 6 , 1 ⟩] / √(37t^2 + 9)

= (36t + t) / √(37t^2 + 9)

= [37t / √(37t^2 + 9)]

---

aN = | r ' (t) x r ' ' (t) | / | r ' (t) |

= | ⟨ 0 , -3 , 18 ⟩ | / √(37t^2 + 9)

= √333 / √(37t^2 + 9)

= √[333 / (37t^2 + 9)]

YES