Q1: For the position vector r = et cos(t) i+ et sin(t) j+ e¯tk, find the velocity and acceleration magnitude at any time of ( t). %3D

Q: Q1. The position vector of a moving point is given as: R(t) = (cos3t) i + (sin3t) j+(3t)k, find (a)…

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Q: 20. r(t) = sin?ti + cos?tj + tan²t k, t= T/4

A: Given :  r(t)=<sin2t, cos2t, tan2t> To find out unit tangent vector , we take derivative of…

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: Q3. For the motion vector r(t) = e cost i + e'sint j %3D find: (i) (ii) ат

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Q: Use the given acceleration vector and initial conditions to find the velocity and position vector.…

A: Given:  Acceleration vector, at = -costi -sintj                      1Initial Vectors,   Velocity…

Q: | The equation 7(t) = (8t) i + (8t – 5t²) j + (7) k gives the position of a particle in space at…

A: Differentiate wrt t for velocity and again differentiate wrt t for acceleration

Q: The velocity of a particle moving in the xy-plane is given by the vector v(t) = (-5 sin(t²), 2e¬6t…

A: we now that if velocity vectoe at time t is  v(t)=vx , vywhere v1  and  v2  are integrable function…

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: A hang glider is spiraling upward along the curve r(t) = (5 sin t, -5cos t, 4t2-3t +1) Find the…

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Q: For the curve defined by 7(t) = (e* cos(t), eʻ sin(t)) find the unit tangent vector, unit normal…

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Q: 21. For the curve 7(t) = sin ti + cos tj, find 7 '(t) and sketch the position vector 7(÷) and the…

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Q: Evaluate e dz where C is the line segment from 0 to 2+2i, parametrized by z(t) = t + it?.

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Q: Find the position vector R(t) given the velocity V(t) = 4e" i+ 4 cos(4t) j+ 12tk and the initial…

A: To find the position vector by the given velocity vector.

Q: Q2/ Find the velocity, speed, unit tangent vector and acceleration of the position vector f(t) at…

A: This is a problem related to vector and derivative of the function. Based on the general formula and…

Q: Q.7: (A) Find the unit vector towards the tangent of the curve y = x at x =1

A: Find the unit vector towards tangent of the curve y=x2 at x=1.

Q: 2. If a particle's position is given by r(t) = <t ₁2 cost, sint), find the IN velocity,…

A: Given: particle position, r→t=t, 2 cos t, sin t We have to find the velocity, acceleration and speed…

Q: Q5: Given r(t) is the position vector in space at time t, find particle velocity and acceleration…

A: Velocity=(et , 4/9 e2t) Acceleration= ( et, 8/9e2t) Direction of motion at t=ln3 is (3, 4)

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: At the given find the moments when the two distance, vectors of velocity and acceleration are…

A: We have the distance vector is R→t=2i→ +5cost j→ +3sint k→ Now, the velocity vector is :…

Q: Find the unit tangent vector T(7) for the line tangent to the space curve r(t) = (12 cost, 12 sint,…

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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: Q3. For the motion vector r(t) = ti+ In (cos t) j, find (i) The velocity (ii) The speed at t=t (iii)…

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Q: 2. Given r(t) = e' cos ti + e' sin tj+ e'k , find the tangential and normal components of…

A: Given: r(t)=et cost i +et sint j+et kTo find the tangential and normal components of…

Q: The position vector of a moving point is given as :R(t)=(cos t)i+(sint) j+(t) k ; the distance…

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Q: For the curve 7 (t) = (F + 1) 7 + (3t – 4) j+ ( – 1)E. Find the unit tangent vector to the curve at…

A: Given:             r→t=t2+1i^+3t-4j^+t2-1k^ Find: v(t)            vt=r't=2ti^+3j^+2tk^ and…

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: Q3. For the motion vector r(t) = ti+ In (cos t) j, find (i) The velocity (ii) The speed at t=T Write…

A: Given motion vector is, rt=t i^ +lncost j^ i To find velocity. vt=ddtrt=ddtt i^ +lncost j^=ddtt i^ +…

Q: Find the position vector R(t) given the velocity V(t) = (4t + 3) i + 6 sin(3t) j+ 6r² k and the…

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Q: Determine the velocity vector of the given path. r(t) = (6 cos?(t), 3t – t3, 2t)

A: We have to find the velocity

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: The path r(t)=(5 sin t) i+(5 cos t) j describes motion on the circle x^2+y^2=25. Find the​…

A: We have to find ​ particle's velocity and acceleration vectors at t=π/3 and π/6 . Given  r(t)=(5 sin…

Q: Q19:- Find the magnitude of the maximum and minimum directional derivatives of the function w =…

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Q: Q3. For the motion vector r(t) = e cost i + e'sint j find: (i) (ii) aT

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Q: Given differential of the function f at the point (3,2) 2. df = 3dr + ddy Find the directional…

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Q: 4 – Let the possition vector is given by 5tk r(t)* = sin3ti+ cost3j¬ + %3D Find a) The velocity b)…

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Q: Let r(t) = (15, cos t, -3 sint) be the position vector of an object. Find the velocity and speed of…

A: Given that rt=15,cost,-3sint  find the velocity and speed of the object

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: r(t) = t²i + 2tj and t = 1, find the tangential and normal components of acceleration. %3D

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Q: The position vector of a moving point is given as :R(t)=(cos t)i+(sint) j+(t) k ; the distance…

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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: Q1/ Find the curve unit tangent vector and length of the indicated portion of the curve r(t) = (9…

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Q: The equation r(t) = i+ 16t j is the position of a particle in space at time t. Find the angle…

A: r(t)=32t i+12t-16t2 jVelocity vector v(t) = drdt=32 i+12-32t j

Q: Q6 Given the following vector-valued function, draw its graph and then calculate its arc length. 7 =…

A: The given vector-valued function is: x=t ; y = 2t2 ; z = 2In cartesian form,y = 2(x)2 y=2x2 and z =…

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: 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: