In Exercises 1 and 2, graph the curves and sketch their velocity and acceleration vectors at the given values of t. Then write a in the form a = a TT + a NN without finding T and N, and find the value of k at the given values of t. 1. r(t) = (4 cos t)i + (sqrt(2) sin t)j,t=0 and pai/4 2. r(t) = (sqrt(3)sec t2i + (sqrt(3) tan t)j,  t = 0

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: Given that the acceleration vector is a (t) = (-9 cos(3t), –9 sin(3t), 5t), the initial velocity is…

A: Introduction: Suppose, the position vector that represents the position of a particle at any moment…

Q: The position of a particle moving in space at time t>=0 is r(t) = 2i + (4 sin t/ 2) j + (3 -…

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

A: For a position vector r(t), the velocity vector of the r(t)  is given by the first derivative of the…

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: If an object follows the path of the two-dimensional vector-valued function p(t)=(16−cos(t),…

A: Given p(t)=(16−cos(t), 0.6πt−0.15t2−3)

Q: Exercise 4. Show that if a has unit speed, i.e., |la'(t)|| = 1, then its velocity and acceleration…

A: We will prove the following result.

Q: find a parametrization c(t) of the curve satisfying the given condition.  y = x 2, c(0) = (3, 9)

A: The given function is y=x2 and c(0) =(3,9) Let x(0) =3 and y(0)= 9 Take x =t +x(0) Thus, x = t+3

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: Find the unit tangent vector T(t) at the point with the given value of the parameter t. r(t) = 4 sin…

A: Given position vector is                                        r(t)=4sint i-4cost j+tant k tangent…

Q: In Exercises 1 and 2, graph the curves and sketch their velocity andacceleration vectors at the…

A: Consider the given curves. 1. rt=4cos ti+2sin tj ;  t=0 and π4 Differentiate the position vector…

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: Find the velocity and position vectors of a particle, given its acceleration and initial conditions…

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Q: In Exercises 5–8, r(t) is the position of a particle in the xy-plane at time t. Find an equation in…

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Q: (a) Find the position vector of a particle that has the given acceleration and the specified initial…

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Q: (a) Find the position vector of a particle that has the given acceleration and the specified initial…

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Q: find a parametrization c(t) of the curve satisfying the given condition.  y = 3x − 4, c(3) = (2, 2)

A: To find the parametrization

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

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Q: At what times in the interval

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Q: In Exercises 9 and 10, find an equation for the line in the xy-plane that is tangent to the curve at…

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Q: Find the velocity and acceleration vectors in terms of u, and ug- de = 5t, where a is a constant r=a…

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

A: We have to find r(t)

Q: Given that the acceleration vector is a (t) -9 cos(-3t), -9 sin(-3t), 5t), the initial velocity is…

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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 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: In Exercises 1 and 2, graph the curves and sketch their velocity and acceleration vectors at the…

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

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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: Without finding T and N, write the acceleration of the motion a(t) to the position vector r(t) =…

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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: A curve has vector equation r(t) = (In(sin t), In(cost),V2 E 0<t< T/2 Find an equation of the…

A: Given curve, rt=lnsint, lncost, 2t Point: -ln22, -ln22, π22 Equation of a plane passing through the…

Q: 1. Given the vector-valued for R(t) = (2 sin 2t-1)i+ 2 cos 2tj+ 3tk. %3D (a) Find the length of the…

A: Given that Rt=2sin2t−1i+2cos2tj+3tk

Q: 2. The position vector of the particle is 1 R = î cos(t) + 3 sin(t) + k log (0 <t < n/2). sin t' (a)…

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Q: In Exercises 31-36, find a parametrization of the tangent line at the point indicated. (lns)i + s- 1…

A: Given,    r(s)=(ln s)i + (s-1)j + (9s)k   at s = 1 Therefore,          r(s)=(ln s, s-1, 9s)⇒…

Q: Given the position vector 7 (t) = of a particle in space at time t. t

A: Given,           The position vector r→t = t+1, t, 1t of a particle in space at time t.

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

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Q: Find the derivative of the vector function r(t) = In(4 – 2) i+ v6+ij+7e¬6i k -

A: The given vector function is  r(t)=ln(4-t2)i+6+tj+7e-6tk The objective is to calculate the…

Q: Find the unit tangent vector T(t) at the point with the given value of the parameter t. r(t) = 4√ti…

A: Given rt=4t i+3t2 j+3t k, t=1

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

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Q: The figure is a section of a conducting rod of radius R1 = 1.30 mm and length L = 11.60 m inside a…

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Q: In exercises 1–18, sketch the curve traced out by the given vector- valued function by hand. 1. r(t)…

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Q: In Exercises 1–7, find the velocity and acceleration vectors in terms of u, and u,. do 1. r = 0 and…

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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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Q: (a) Find the position vector of a particle that has the given acceleration and the specified initial…

A: To find the position vector of a particle with given acceleration vector and initial conditions

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

A: Topic :- Calculus

Q: In Exercises 1–3, find the curve’s unit tangent vector. Also, find the length of the indicated…

A: a) Find r’(t).

Q: In Exercises 5–8, r(t) is the position of a particle in the xy-plane at time t. Find an equation in…

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Q: C) If R(t) = (In(t2 + 1)i + (tan-t)j + (VE +1)k, is the position vector of a particle at time (t).…

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Vector Arithmetic

Vectors are those objects which have a magnitude along with the direction. In vector arithmetic, we will see how arithmetic operators like addition and multiplication are used on any two vectors. Arithmetic in basic means dealing with numbers. Here, magnitude means the length or the size of an object. The notation used is the arrow over the head of the vector indicating its direction.

Vector Calculus

Vector calculus is an important branch of mathematics and it relates two important branches of mathematics namely vector and calculus.