Compute the unit binormal vector and torsion of the curve. r(t) = (11t, 5 cos t, 5 sin t) (5, 11 sin t, - 11 cos t) V146 11 O A. B(t) = 146 (0, 11 cos t, - 11 sin t) V146 1 O B. B(t) = 146 (0, - 11 cos t, - 11 sin t) V146 11 OC. B(t) = 146 (5, - 11 sin t, 11 cos t) V 146 11 O D. B(t) = 146

Q: In Exercises 1–4, find the curve’s unit tangent vector. Also, find the length of the indicated…

A: Since you have asked multiple questions, we will solve the first question for you. If you want any…

Q: What is the unit tangent vector to the curve r(r)=sint i+2cost j when t=π/4

A:

Q: Q1/Find the tangent and normal vector for curve = (5sin n i+5 cos n j+n k)

A: we have to find the tangent and normal vactor

Q: Consider the helix r(t)=(cos(3t), sin(3t), 3t). Compute, at t=π/6 The unit normal vector N

A: The given vector of helix is,

Q: Q2. A particle is moving along the curve whose position vector is given as R(1) = (1) i + ( In…

A: Given

Q: Find the tangent, normal, and binormal vector functions for the following three curves:   curve 1:…

A:

Q: The position vector for a particle moving on a helix is  c(t) = (2 cos(t), 4 sin(t), t2).   Find a…

A: The position vector of the particle moving on a helix is given as follows.

Q: 2. Find the unit tangent vector for the curve r(t) = (t cost, tsin t).

A:

Q: 4.a. Find the unit tangent and normal vectors to r(t) = 4 cost i + sin t j at t = /2. Sketch a…

A: Given that rt=4cost i +sin t j Then r't=-4sin t i  +cos t. The formula for tangent vector is…

Q: /Find the tangent and normal vector for curve r= (5sin n i+5 cos n j+n k) n=(In 2t)

A:

Q: Show that at every point on the curve r (t) = ⟨et cos t, et sin t, et⟩ the angle between the unit…

A:

Q: Find the unit tangent vector of the given curve. r(t) = (7t cost-7 sin t)j + (7t sin t + 7 cost) k…

A:

Q: The principal unit normal vector on the curve r(t)= (2 sint,2cost) %3D (a) always points directly…

A:

Q: Sketch the curve parametrized by r(f) = (1 – 12, 1) for -1sı<1. Compute the tangent vector at i = 1…

A: Let's first draw the curve -  Domain - -1 ≤ t ≤ 1 where, we can take values of t as -  t = -1, -0.5,…

Q: Q1/Find the tangent and normal vector for curve (5sin n i+5 cos n j+n k)

A: We have given r(t) and we have to find tangent and normal vector to r(t)

Q: a) By first sketching its projection onto the yz-plane, sketch the curve r(t) = (t, cos(t), 2…

A: For (a), We will plot the sketch of r(t)=t, cos(t), 2 sin(t),  0≤t≤4π onto the yz-plane (that means…

Q: Sketch the curve represented by the vector-valued function                               r(t) = 2…

A: Given information: Vector-valued function rt=2costi+tj+2sintk trigonometric identity, sin2t+cos2t=1…

Q: Find the unit tangent vector to the curve at the specified value of the paramet r(t) = 4 cos(t)i + 4…

A:

Q: Show that at every point on the curver(t) = ⟨et cos t, et sin t, et ⟩ the angle between the unit…

A:

Q: Find a tangent vector at t=a for the following parameterized curve. r(t) = (6 sin t,5 cost, cos -…

A: Given the parametrized curve r(t) :

Q: Find a tangent vector at t=, for the following parameterized curve. 2 r(t) = (t, sin 6t,7 cos t)…

A:

Q: Find the Tangent vector, the Normal vector, and the Binormal vector (T, N and B) for the curve F(t)…

A:

Q: Find the unit tangent vector to the curve defined by 7(t) = (3 cos(t), 3 sin(t), 4 sin (t)) at t =…

A:

Q: Find the unit tangent vector T, the unit normal vector N, and the binormal vector B for curve r at…

A:

Q: Let the velocity vector be v(t) = (2 cos(t), 5t“, 12 exp(3t)) and the initial position vector be…

A:

Q: Find a tangent vector at t= for the following parameterized curve. r(t) = (5t, sin 6t,3 cos t)…

A: Recall: Tangent vector to the given curve r(t)  is r'(t).

Q: Find the unit tangent vector of the given curve. r(t) = (6 sin 39t) i + (6 cos ³9t)j O A. T(t) = (6…

A:

Q: for the follawing , find the unit tangent vector: parameterized curve rit) =<2 sint) , 2 cos lt),5…

A: Given: r(t)=2sin(t),  2cos(t), 5cos(t),  0≤t≤π The unit tangent vector to curve r(t) is…

Q: Find the Tangent vector, the Normal vector, and the Binormal vector (T, N and B) for the curve r(t)…

A: This question can be solved using the concept of vectors and derivatives.

Q: An object moving along a curve in the xy-plane has velocity vector v(t) =. At time t = 0, the object…

A:

Q: Consider the vector-valued function r(t) = (sin 2t + cos 2t)i + (sin 2t – cos 2t)j. (a) Find the…

A:

Q: Find the position vector for a particle with acceleration, initial velocity, and initial position…

A: Finding the Position Vector for a Particle: The integration of the acceleration vector is the…

Q: Find a vector parametrization of the circle of radius 5 in the xy-plane, centered at the origin,…

A: Solution: Given, Circle having center at (0, 0) and radius is 5 and

Q: Sketch the curve with the given vector equation. Indicate with an arrow the direction in which t…

A:

Q: 3. A particle moving along a curve in the xy-plane has position (x(t1),y(t)), with x(1)= 2t +3sin t…

A: Velocity is equal to the first derivative of position vector   Let x =f1t and y =f2t then velocity…

Q: Find the position vector for a particle with acceleration, initial velocity, and initial position…

A:

Q: A particle is moving along the curve whose position vector is given as R(t) = (t)i + ( În (cost)) j…

A:

Q: 9- The unit tangent vector T(t) = (- sin ti + cos tj+3k) is given. Using definition of unit normal…

A: The general form of the parametric equation of the curve in three dimensional space is given by ,…

Q: 8. Find the curvature of the vector-valued function r(t) = at the point t = -1.

A: given function is rt=3t2,-4t,1t find the curvature at t=-1 k=r→'t×r→''tr→'t3

Q: Compute the unit binormal vector and torsion of the curve. r(t) = (10t, 3 cost, 3 sin t) (3, 10…

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Q: Find the unit tangent vector to the curve at the specified value of the parameter. (-)- II r(t) = 3…

A:

Q: Compute the unit binormal vector and torsion of the curve r(t) = (8t, 5 cost, 5 sint (5, 8 sint, -8…

A:

Q: Find the position vector for the particle with acceleration, initial velocity, and initial postion…

A:

Q: 2. Let C be a curve given by r(t) = (e' sin t, e' cos t, /2e'), t is real. (a) Find the binormal…

A: Given, r(t)=etsin t, etcos t,2et

Q: A particle moving along a curve in the xy-plane has position (x(t), y(t)) = (-9 sin(9t), 3t² + 10t),…

A: Solved below

Q: Find the unit tangent vector to the curve r(t) = (2sin(t),2cos(t)) at the points t =-n, t=0 and t…

A:

Q: Let the velocity vector be v(t) = (4 cos(t), 5t“, 4 exp(2t)) and the initial position vector be r(0)…

A:

Q: Let r(t) = (sin(2t), 3t, cos(2t)), t E [-T, "] be the position vector of a particle at time t. (a)…

A: I am going to solve the given problem by using some simple calculus to get the required result.

Arc Length

Arc length can be thought of as the distance you would travel if you walked along the path of a curve. Arc length is used in a wide range of real applications. We might be interested in knowing how far a rocket travels if it is launched along a parabolic path. Alternatively, if a curve on a map represents a road, we might want to know how far we need to drive to get to our destination. The distance between two points along a curve is known as arc length. 

Line Integral

A line integral is one of the important topics that are discussed in the calculus syllabus. When we have a function that we want to integrate, and we evaluate the function alongside a curve, we define it as a line integral. Evaluation of a function along a curve is very important in mathematics. Usually, by a line integral, we compute the area of the function along the curve. This integral is also known as curvilinear, curve, or path integral in short. If line integrals are to be calculated in the complex plane, then the term contour integral can be used as well.     

Triple Integral

Examples: