In: Math

Consider the following vector function. r(t) =<3t, 1/2 t2, t2> (a) Find the unit tangent and unit normal vectors T(t) and N(t).

(b). Find the curvature k(t).

Consider the helix
r(t)=(cos(2t),sin(2t),−3t)r(t)=(cos(2t),sin(2t),−3t).
Compute, at t=π/6
A. The unit tangent vector T=T= ( , , )
B. The unit normal vector N=N= ( , , )
C. The unit binormal vector B=B= ( , , )
D. The curvature κ=κ=

15.
a. Find the unit tangent vector T(1) at time t=1 for the space
curve r(t)=〈t3 +3t, t2 +1, 3t+4〉.
b. Compute the length of the space curve r(t) = 〈sin t, t, cos
t〉 with 0 ≤ t ≤ 6.

(1 point) For the given position vectors r(t)r(t) compute the
unit tangent vector T(t)T(t) for the given value of tt .
A) Let r(t)=〈cos5t,sin5t〉
Then T(π4)〈
B) Let r(t)=〈t^2,t^3〉
Then T(4)=〈
C) Let r(t)=e^(5t)i+e^(−4t)j+tk
Then T(−5)=

compute the unit tangent vector T and the principal normal unit
vector N of the space curve R(t)=<2t, t^2, 1/3t^3> at the
point when t=1. Then find its length over the domain [0,2]

Given the vector function r(t)=〈√t , 1/(t-1) ,e^2t 〉 a) Find: ∫
r(t)dt b) Calculate the definite integral of r(t) for 2 ≤ t ≤ 3
can you please provide a Matlab code?

Consider the vector function given below.
r(t) =
2t, 3 cos(t), 3 sin(t)
(a) Find the unit tangent and unit normal vectors T(t) and
N(t).
T(t) =
N(t) =
(b) Use this formula to find the curvature.
κ(t) =

Show the complete solution.
Determine the unit tangent vector (T), the unit normal vector
(N), and the curvature of ?(?) = 2? ? + ?^2 ? – 1/3 ?^3 k at t =
1.

FOR THE PARAMETRIZED PATH r(t)=
e^tcos(πt)i+e^tsin(πt)j+e^tk
a) find the velocity vector, the unit
tangent vector and the arc lenght between t=0 and t=1
b) find a point where the path given by r(t)
intersects the plane x-y=0 and determine the angle of intersection
between the tangent vector to the curve and the normal vector to
the plane.

Integration of the vector function: F(t)
=et.sin2ti +
t2j +
t2.e2tk

Find the vectors T, N, and
B at the given point.
r(t) = < t2, (2/3) (t3), t > , <
4, -16/3, -2>

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