Q.1: Express the vector 2i + 3j – k in the primed triad i'j'k' in which the x'y'- axes are rotated about the z-axis (which coincides with the z'-axis) through an angle of 30°.

Q: 4. Find the divergence and curl of the following vector function: F(x,y, z) = ( arctan(ry),…

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Q: 2.10 (a) Express the vector field H = xy'za, + x'yza, + xyz'a. in cylindrical and spherical…

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Q: 4. For any time t 2 0, if the position of a particle in the xy-plane is given by x = r² +1 and y=…

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Q: 9. An object moving along a curve in the xy-plane has position (x(t),y(t)) at dy dx :=1+ tan(r²) and…

A: According to Bartleby q&a guidelines I will attempt only one question

Q: 3. Let the curve C be traced by the vector-valued function R(t) = cos(at) î + at j+ sin(rt) k. %3D…

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Q: 3. Consider the following helix with vector equation 7 (t) = 2 cos (t) 7 (t) 7 + tỉ + 2 sin (a)…

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Q: 1. Find the unit tangent vector T(t) and principal normal vector N(t) for the curve r(t) = (2…

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Q: 6. Determine the values oft between 0 and 27 such that the tangent vector to the cycloid r(t) = (t –…

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Q: 6: Find the work done by the force F = -7y i+ 10x j along one loop of the curver = sin 70 (in the…

A: Given

Q: Express the following as a vector valued function 7 (t). 1. The line L defined by y+3=3x and z+1 =0…

A: We will express the line L defined by y+3=3x and z+1=0 hyperbola -x2-2x+2y2+4y=-2 as a vector valued…

Q: At time to, a moving particle has the velocity vector v = 7i and the acceleration vector a = 7i +…

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Q: 2. Suppose that a motion described by the vector valued function (i.e. position function) r(t) has…

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Q: 5. Suppose a particle is moving along a smooth path r(t) with constant speed. If a(t) + 0, show that…

A: Two vectors is perpendicular if their dot product is zero.

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: give the position vectors of particles moving alongvarious curves in the xy-plane. In each case,…

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Q: Find the vector equation of the tangent line to F(t) = (2 sin 2t, cos 2t, – 6t) at the point where…

A: Consider the given vector. rt=2sint,cos2t,-6t   Find the derivative of the vector with respect to t.…

Q: Q1- The divergence of the function f= (r cos 0)e, in spherical coordinates is..........

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Q: 3. Consider the curve C defined by the vector-valued function R(t) = -2):+-3)j+ cos(t – 2)k, - oo…

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Q: Q1- The divergence of the function f = (r5 cos 0)e, in spherical coordinates is...

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Q: 3. A unit vector tangent to the curve y x2 at the point (2, 4).

A: We are entitled to solve only one question at a time so we will answer only the first question.…

Q: 3. Let the curve C be traced by the vector-valued function R(t) = cos(zt) î + at j+ sin(rt) k. %3D…

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Q: 6. () Let r(ø, 0) = a sin ø cos đi+a sin ø sin 0j+a cos øk be the vector equa- tion of a sphere of…

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Q: 18. Determine the values oft between 0 and 27 such that the tangent vec- tor to the cycloid r(t) =…

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Q: Find the unit tangent vector to the curve defined by 7 r(t) = (5 cos(t), 5 sin(t), 2 sin² (t)) at t…

A: To find the unit tangent vector to the curve defined by r→t=5 cost, 5 sint, 2 sin2t

Q: Compute the principal unit tangent vector at t = T/2. F(t) = sin(2t) î + 2 cos(t) ĵ + tk COS 2 2 - -…

A: unit tangent vector

Q: 2.1 Given that: f = (i Cos 8 + j Sin 0 ) is a unit vector in the direction of f and that 6 = (- i…

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Q: 2. Suppose that the position of a particle is r(t) = (cos(nt), ² sin(nt), t), 0≤t≤ 2n. A. Calculate…

A: Understanding of derivative while finding velocity and acceleration from position vector is…

Q: 3. A particle moves in the xy-plane so that at any time t, its coordinates are given by x = t° -1…

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Q: (1) Vf (f is a scalar function) 2) ř(t) and r(s), t is time and s is arch length.

A: This is a problem of vector calculus.

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

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Q: Compute the tangent vector to the given path. c(t) = (4t sin(t), 6t) %3D

A: The tangent vector for a vector r→t is r→'t.

Q: 10. A particle moves in the xy-plane so that the position of the particle is given by x(1) =1+cos t…

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Q: 17. Determine the value of t between 0 and 27 such that the tangent vector to the cycloid r(t) = (1…

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Q: 8. Suppose that over a region of space the electrical potential V is given by V(x,y,z) = 5x² – 3xy +…

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Q: 3. Find the work of the vector field F(2, y, ) = yi along the curve which is obtained as the…

A: To find the work of the vector field Fx,y,z=yi along the curve obtained by the intersection of…

Q: V3 t², a cost + at sin t) 2 r(t) = (a sint – at cos t,

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Q: The integral of vector A= 0.25p cos(q)ap +2zaz Over the bottom of cylinder specified by 1<Zs2, p = 2…

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Q: 4y The divergence of the vector G = i+ sin y j+ 3k is :- -8y + cosy x3 الخيار 2 O

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Q: 16. Sketch the cycloid r(t) = (t- sint, 1 - cos t) together with its tangent vectors at t = and .…

A: Solved the above problem.

Q: A particle moves through 3-space in such a way that its acceleration is a(t) = 8sin 2ti +8 cos…

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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: Suppose that C is the graph of a smooth vector-valued function r(s) parametrized by arc length with…

A: Introduction: The tangent vector to a vector valued function rt is formulated by the equation r't…

Q: Q1- The divergence of the function f = (r5 cos 0)e, in spherical coordinates is.. .

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Q: Q1- The divergence of the function f (r5 cos 0)e, in spherical coordinates is...

A: The given problem is to find the divergence of the vector function in spherical coordinates.

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: 5) Integrate the vector field h a,y)= y2i+ (2xy-e*)J C parumetrizal by t€ £0, 5 3. over the circular…

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Q: 1. Calculate the work done when a force F = 3xyî – y?j moves a particle in the xy-plane from (0,0)…

A: Take dot product along limit of integration and integrate

Q: 4. If a particle moves in the xy-plane so that at time t its position vector is sin 31 ,3r2 find the…

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Q: 4. Given the vector field Č = p² sin o a, + (z² cos – 1)as + (p² – z)a, Determine the x-component of…

A: Given the vector field