Q1/ Find the curve unit tangent vector and length of the indicated portion of the curve r(t) = (9 sin 2t)i + (6cos 2t)j + 5t k

Q: 1. Calculate the moment of inertia about x-axis of the first quadrant portion of the curve: y = 4-…

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Q: Find the unit tangent vector to the curve defined by F(t) = (4 cos(t), 4 sin(t), 3 sin²(t)) at t = 3…

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

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Q: Find the slope of the tangent line to the parametric curve [x = 4 cos(t) y = 3 sin(t) at the point

A: A detailed solution is given below.

Q: A unit tangent vector to the curve r(t) = [sin(t), cos(t), 2f] at the point A:(0, — 1, 2п) is given…

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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: Q: Find the equation of the tangent plane to the surface:- z-Ln(xj) -x5J , atthe poit (7,4)

A: The given surface is z=lnx2y-xy=fx,y at the point -12, 4. The equation of tangent plane is…

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: What is the unit tangent vector to the curve r(t)=⟨cost, sint, et⟩ at the point where t=0?

A: Given:          r(t)=cost,sint,et at t=0           r→(0)=cos(0),sin(0),e0          r→(0)=1,0,1…

Q: find the curve’s unit tangent vector. Also, find thelength of the indicated portion of the curve. .…

A: Given:                  rt=cos3tj+sin3tk,     0≤t≤π2                 v=ddtrt=ddtcos3t j+sin3t…

Q: 4. Find an equation of the tangent plane to the surface x = u? + 1,y = v³ + 1,z = u + v at the point…

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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: ) Find the orthogonal trajectories of the family of curves r = 3C sin 0, where C being parameter.

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Q: Find the unit tangent vector of the curve: 7(t) = (3 cos t , 3 sint, t2) at t = 0. %3D

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Q: The parametric equations of a curve are given by x = In (VE +1) and y = cosh-' (2t). dy when t = 1.…

A: Given x=lnt+1 and y=cosh−12t We have to find the dydx when t=1 We know that dydx=dydtdxdt.....1…

Q: At an air show, a jet has a trajectory following the curve y = x2. Ifwhen the jet is at the point…

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Q: 7) Find the unit tangent vector for the parameterized curve: r(t) = cos ti+ sin tj+ sin tk, 0≤t< 2π

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Q: Find a parametrization of the tangent line at the point indicated. r(t) =〈cos 2t,sin 3t〉, t = π/4

A: Given :  r(t) =〈cos 2t,sin 3t〉, t = π/4

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

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Q: Find an equation of the tangent plane to the surface at the given point. 2 x2 + 2z?ey - x = 33, P =…

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Q: 2. Find thé moving trihedral of the curve C traced by the vector-valued function R(t) 4 cost i+4…

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Q: Find the equation of the tangent line to the parametric curve at (1,3). 2=1+ Int y =t² +2

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Q: Find the tangent line(s) to the parametric curve given by a = t5 – 4t3 y = t? at (0, 4).

A: Find the tangent line(s) to the parametric curve given by x = t^5-4t^3 , y = t^2 at (0,4).

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: Find the angle between the radius vector to the curve r = 2a sin 3u and its tangent when u = pai/6.

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Q: (5) Find the equation of the tangent plane to z = 1² + xy +3y* at r = 1, y = 1.

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Q: Find the unit vector towards the tangent of the curve y = x at x =1

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Q: Find the equation of the tangent line to the parametric curve at the indicated point. Show all work.…

A: Equation of tangent line : y - y0 = m ( x - x0 ) m is the slope of tangent line .

Q: The parametric equations of a curve are given by x = In (VE +1) and Vt + y = cosh-(2t). X = dy when…

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Q: Find an equation of the tangent plane to: 3x +y = z at P(-1,1,-2) %3D

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

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Q: (11) Find the tangent line(s) to the parametric curve given by * = t5 - 4t3 y = t2 at (0, 4)

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Q: Find an equation of the tangent(s) to the curve with paramnetric equations ¤ = 1+ VE, y = e" at the…

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Q: Find equations for the a) tangent line and b) normal plane to the curve 1 x=t- cost, y = 3+ sin 2t,…

A: x=t-cost, y=3+sin2t, z=1+cos3t We know that rt=xt, yt, zt dxdt=1+sint, dydt=2cos2t, dzdt=-3sin3t At…

Q: 13. Find the point on the curve 7(t) = (2 sin t, 2 cos t, t), 0<ts 7, where the tangent line is…

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Q: Find the slope of then tangent line to the parametric curve t t2 + 25, y 6. at t = -1 and t = 1…

A: Shown below

Q: Find an equation of the plane tangent to the following surface at the given point. 1 sin (xyz) =

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

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Q: How do you find the Unit Tangent Vector to a curve C when you are given x(t) and y(t)? Where you are…

A: Suppose we are given a smooth curve C:r(t)→=&lt;x(t),y(t)&gt;on an openinterval I. Then a unit…

Q: Find the length of the curve x = 8 cos t + 8t sint,y= 8 sin t – 8t cos t ; 0 < t %3D -

A: The length of the parametric curve is given by  L=∫abx't2+y't2dt.

Q: x = 1+ vt, y = et X =1+1 Y. %3D

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Q: The tangent vector to a curve r is given by r′(t)=⟨16t,128t^2,1⟩. Let T be the unit tangent vector.…

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Q: Find the length of parametrized curve given by  x(t) = 6t2 + 12t, y(t) = -4t3- 12t2 - 9t where t…

A: Consider the parametric curve given by: x(t)=6t2+12t and y(t)=-4t3-12t2-9t for t from 0 to 1. The…

Q: We start by finding the tangent vector to the curve. For r(t) = (10t, 3 cos(t), 3 sin(t)), we have…

A: Given rt=10t ,3cost ,3sint

Q: Find the lengthof the curve r= a sin (0/2), 0s 0S T, a>0

A: The formula for finding arc length is L = ∫0πr2+ drdθ2dθ, where r = a sin2(θ2)

Q: Find the orthogonal trajectories of the family of curves r 5C sin 0, where C being parameter.

A: Follow the below steps to find the orthogonal trajectories. Differentiate the function with respect…

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.