Well, by definition, the tangent vector is the vector that is the result of differentiating every component of the function. Hence, the tangent vector is T=(3,-5sint,5cost). This has a magnitude of [tex]|r|^2=3^2+(-5sint)^2+(5cost)^2[/tex] which yields |r|=[tex] \sqrt{34} [/tex] (by using the known trigonometric formula [tex]cost^2+sint^2=1 [/tex] that holds for all t). Thus, the unit tangent vector t is given by: t=[tex] \frac{T}{\sqrt{34}} [/tex]. Now, to find a normal unit vector, we just need to find a normal vector N and scale it appropriately. To find a normal unit vector, we can just find vector that satisfies V*T=0 (dot product of vectors). By inspection, we see that the vector N=(0, cost, sint) satisfies this requirement and that it is also a unit vector (due to the aforementioned trigonometric identity). Hence, a normal unit vector to our vector function is N.
