Given that r (t) = (5t cos 2t) j- (5t sin 2t) k, is the position vector of a moving particle, find the following quantities: 1. The velocity, speed, acceleration, and curvature of the particle, 2. The tangential and normal components of acceleration of the particle.

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

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Q: Q2:- A/ Given a position vector: R (t)= 3 cos ti+3 sin tj+ t2 k find 1- The velocity vector and…

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Q: The position vector of a particle moving along a helix is c(t) = (cos(t), sin(t), t²). (a) Find the…

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Q: Given that the acceleration vector is a (t) = (-16 cos(-4t), – 16 sin(-4t), –t), the initial…

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Q: Find the velocity and acceleration vectors in terms of u, and ue en r=6 sin t and 0= 9t

A: The velocity is given by            v = r' ur +rθ' uθandThe accerleration is given by        a =r"-r…

Q: 13. r(t) = ei+e j+4k at t 0

A: Given vector :

Q: Find the unit tangent vector and the length of the indicated portion of the curve: (a) r(t) = (6 sin…

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

A: Topic - vectors

Q: Given the vector function *(t) = (t, cos (2t)) Calculate the curvature when t=5.4

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Q: Find the velocity and acceleration vectors in terms of u, and ug. r= a(9 + sin t) and 0 = 3 - e',…

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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: 1. Suppose that r(t) = (3 cos(3t) t,3 sin(3t), 5t). a. Determine the velocity, the acceleration and…

A: For velocity vector diffrenciate displacement vector. And for acceleration vector diffrenciate…

Q: 1. Suppose that r(t) = (3 cos(3t) · t, 3 sin(3t), 5t). a. Determine the velocity, the acceleration…

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Q: How would I find the unit tangent vector T(t) at the point with the given value of the parameter t.

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Q: a) The position of point moving in space is given by R(t) = 3 tan t i + cott j +2 lnt k, find the…

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Q: Find the length of the curve with the given vector equation. r(t) = cos ti+ sin tj+ 3t k; 0sts3

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Q: Find the unit tangent vector and the parametric equation of the line tangent to the given curve at…

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Q: At what times in the interval

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Q: An object is spinning at a constant speed on the end of a string, according to the position vector…

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

A: Given :  The the vector r (t) = 6t i→ - t2 j→   and the parameter t = 2.   To determine : The unit…

Q: 1. Consider the position vector r(t) = (t³) i + (3t) j + 4k and at t = 1 find the: (a) curvature,…

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

Q: For time 1 > 0, the position of a particle moving in the xy-plane is given by the parametric…

A: Velocity==rate of change of distance per unit time=displacementtime=dsdt. Given that: x=4t+t2  and…

Q: Find the unit tangent vector T(t) at the point with the given value of the parameter t. 2t r(t) =…

A: In this question, we want to find the tangent unit vector of r(t) at t=0.

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

A: We have to find r(t)

Q: b) Find the unit tangent vector for.. r(t) = at t=0

A: The unit tangent vector can be calculated using the formula T→(t)=r→′(t)‖r→′(t)‖. The denominator is…

Q: Find the unit tangent vector of the given curve. r(t) = 12t5i - 4t5j + 3t°k %3D

A: Given the vector function, r(t) , we call r′(t) the tangent vector provided it exists and provided…

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

A: Here,  a→(t)=3t i^+6 sin(t)j^+cos(2t)k^⇒v→(t)=∫3t i^+6 sin(t)j^+cos(2t)k^dt⇒v→(t)=3t22 i^+6…

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

A: Given,

Q: B. Given that r (t) = (3t cos t) j - (3t sint) k , is the position vector of a moving particle, find…

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

A: We have acceleration, initial velocity, and initial position, and if we integrate acceleration with…

Q: Consider the following 3D trajectory: t p(t) = sin(2t) i +3 cos(t) j+ k 2n a. Calculate the position…

A: In geometry and physics and its engineering applications, we use two kinds of quantities, scalars,…

Q: Find the (a) velocity vector, (b) speed, and (c) acceleration vector for the particle that moves…

A: rt=2sint2 i+2cost2 j a). Velocity vector is the derivative of position vector. That is,…

Q: 8. Given the vector r(t) = (sint,cos?, In(cos?)), find the equation of the tangent line in…

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Q: Find the acceleration vector for the position vector r(t): = (sin(11t), 2t", e-10t).

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Q: Find the velocity and acceleration vectors in terms of u, and ug. r= a(6 + cos t) and 0= 5+ e where…

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Q: Given the vector function r(t)=(1−t³, 3t+1, 4+1²), the equation of the tangent parabola/quadratic to…

A: We have to solve given problems:

Q: Find the velocity and acceleration vectors in terms of u, and ug. de r= a(7 - cos 0) and = 6, where…

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Q: For the position vector R(1) = e'i+e²ª j, the velocity at t=ln3 is:

A: Consider the given position vector: R→t=eti+29e2tj Find Velocity vector: Differentiate w.r.t "t",…

Q: Find the unit tangent vector to the curve at the specified value of the parameter. r(t) = ti + 8t2j,…

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Q: Find the velocity and acceleration vectors in terms of u, and ue. de = 4t, where a is a constant dt…

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Q: Q1: For the position vector r = et cos(t) i+ et sin(t) j+ e¯tk, find the velocity and acceleration…

A: Note : There are many questions in the above image. So I will answer the first question and…

Q: Q1: For the position vector r = et cos(t) i + et sin(t) j+ etk, find the velocity and acceleration…

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Q: Find the unit tangent vector to the curve at t r(t) = t°i + 3t?j, t= 2 T(2) =

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Q: The position vector in the xy plane is [t2,sin(t)].  What is the distance traveled by the particle…

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Q: 3. For the motion vector r(t) = ti+ In (cos t) j, find (i) The velocity (ii) The speed at t=N Write…

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Q: Parametrize the given curve. 4x + 6y2 = 3 c(t) =

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Q: В. Given that r (t) = (3t cos t) j – (3t sin t) k , is the position %3D vector of a moving particle,…

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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: