Need help. What is the answer? Thank you sir.

Question

The parametric equations of a curve is r(t)=2i+tj+t²k.

a) sketch the curve r(t) for -3≤t≤3.

b) Find the followings when t=0

i) The unit tangent vector, T.

ii) The principal unit normal vector, N.

Answer 1

when t=0, x=2, y=z=0

r(0)= <2,0,0> magnitude = 2, direction along the x axis, line segment from (0,0,0) to (2,0,0)

unit normal vector: <1,0,0> magnitude=1, in the same direction,along the x axis, from the origin to (1,0,0)

x=2, y=t, z=t^2 = y^2'

t=-3 <2,-3,9>' r(-3) has magnitude sqr(4+9+81) = sqr94

t=3 <2,3,9> r(3) has same magnitude = sqr94

since x is a constant =2 regardless of t value

picture the graph in 2 dimensions, the y z dimensions

sort of like just a 2 D curve where y=x^2 but this is z=y^2

so the graph is just the same parabolic curve