Question

Given r(t) = (e^{t} cos(t) )i + (e^{t} sin(t) )j
+ 2k. Find

(i) unit tangent vector T.

(ii) principal unit normal vector N.

Answer #1

Unit tangent vector and principal normal vector

Given r(t)=sin(t)i+cos(t)j−ln(cos(t))k, find the unit normal
vector N(t) evaluated at t=0,N(0).

Find a unit tangent vector to the curve r = 3 cos 3t
i + 3 sin 2t j at t =
π/6 .

Find the unit tangent vector T and the principle unit normal
vector N of ⃗r(t) = cos t⃗i + sin t⃗j + ln(cos t)⃗k at t = π .

Consider the following vector function.
r(t) =
6t2, sin(t) − t cos(t), cos(t) + t sin(t)
, t > 0
(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) =

Find an arc length parametrization of
r(t) =
(et
sin(t),
et
cos(t),
10et )

Find r(t) for the given
conditions.
r''(t) = −7
cos(t)j − 3
sin(t)k, r'(0)
=
3k, r(0)
= 7j

17.)Find the curvature of r(t) at the
point (1, 0, 0).
r(t) =
et
cos(t),
et
sin(t), 3t
κ =

Find the unit tangent vector T and the principal unit normal
vector N for the following curve.
r(t) = (9t,9ln(cost)) for -(pi/2) < t < pi/2

Given that the acceleration vector is a(t)=(-9 cos(3t))i+(-9
sin(3t))j+(-5t)k, the initial velocity is v(0)=i+k, and the initial
position vector is r(0)=i+j+k, compute:
A. The velocity vector v(t)
B. The position vector r(t)

20. Find the unit tangent vector T(t) and then use it to find a
set of parametric equations for the line tangent to the space curve
given below at the given point.
r(t)= -5t i+ 2t^2 j+3tk, t=5

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