Consider the vector function given below. r(t) = 3t, 5 cos(t), 5 sin(t) (a) find the unit tangent and unit normal vectors t(t) and n(t).'

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 which yields |r|= (by using the known trigonometric formula that holds for all t). Thus, the unit tangent vector t is given by: t=. 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.