Question

At a given point on a smooth space curve r(t), T(t) is the unit tangent vector, N(t) is the principle unit normal vector and B(t) is the binormal vector. Which of the following are correct? (The multiple-choice question might have more than one correct answer. Circle all correct answers for full credit.) Group of answer choices

A)None of the above has to be true.

B) T ( t ) ⋅ T ′ ( t ) = 0

C) | B ( t ) | = 1

D) T(t)x N(t)=0

Answer #1

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 .

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

Find the unit tangent vector T(t) and the curvature κ(t) for the
curve r(t) = <6t^3 , t, −3t^2 >.

6. Given vector function r(t) = t2 − 2t, 1 + 3t, 1 3 t 3 + 1 2 t
2 i (a) Find r 0 (t) (b) Find the unit tangent vector to the space
curve of r(t) at t = 3. (c) Find the vector equation of the tangent
line to the curve at t = 3

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

Consider the following vector function.
r(t) = <9t,1/2(t)2,t2>
(a) Find the unit tangent and unit normal vectors
T(t) and
N(t).
(b) Use this formula to find the curvature.
κ(t) =

Given r(t) = (et cos(t) )i + (et sin(t) )j
+ 2k. Find
(i) unit tangent vector T.
(ii) principal unit normal vector N.

15. Find the principle unit normal vector to the curve given
below at the specified point.
r(t)= t i + 4/t j, t=2

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(t) at
the point with the given value of the parameter t.
r(t) =
t2 − 4t, 1 + 5t,
1
3
t3 +
1
2
t2
, t = 5

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