In: Math

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.

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]

(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)=

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.

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).

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.

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=

E(y,t) =
Eocos(ky-ωt)x
where x is the unit vector in the x direction,
is a solution of your derived equation. In what direction is this
wave propagating?
Suppose the wavelength of the light is 700nm, what is k in rad
nm-1? What is ω in rad/s? If Eo=300V/m, what is the
electric field at time t=3.5fs (femto seconds) and y = 300nm? What
is the force (magnitude and direction) this field exerts on an
electron?
Bonus: What is the corresponding...

For a position vector
d=(-7.1i+-0.1j)m,
determine the x-component of the unit vector.
For a position vector
d=(-7.1i+-0.1j)m,
determine the y-component of the unit vector.
The direction of a 454.7 lb force is defined by the unit vector
u1=cos(30°)i+sin(30°)j.
Determine the x-component of the force vector.

Show for the following recurrence that T(n) = T(n/3) + n*log(n)
is O(n*log(n)) (not using the Master theorem please)

Assume that a 3D polygon is tranformed by a transformation
matrix T. Assume the normal vector of the polygon is n. Can we use
the formula Tn to compute the normal vector of the transformed
polygon? If not, drive the correct formula.

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