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 = -3, r(-3) = 2i - 3j + 9k

When t = -2, r(-2) = 2i - 2j + 4k

When t = -1, r(-1) = 2i - j + k

When t = 0, r(0) = 2i + 0j + 0k

When t = 1, r(1) = 2i + j + k

When t = 2, r(2) = 2i + 2j + 4k

When t = 3, r(3) = 2i + 3j + 9k

Hopefully you can be able to sketch from here.

The unit tangent vector is calculated by the formula T(t) = r'(t)/||r'(t)|| where ||r'(t)|| represents the magnitude of r'(t):

T(t) = r'(t)/||r'(t)||

T(t) = (0i+1j+2tk)/√[0²+1²+(2t)²]

T(t) = (0i+1j+2tk)/√(1+4t²)

At t=0:

T(0) = (0i+1j+2(0)k)/√(1+4(0)²) = j

The principal unit normal vector follows a similar formula, which is N(t) = T'(t)/||T'(t)||:

N(t) = T'(t)/||T'(t)||

N(t) = (0i+0j+2k)/√(0²+0²+2²)

N(t) = (0i+0j+2k)/√(4)

N(t) = (0i+0j+2k)/2

N(t) = 0i+0j+1k

At t=0

N(0) = k

Hope this helped!