Find T, N, and k for the space curve, where t> 0. r(t) = ( cos t+t sin t)i + ( sin t-t cos t)j + 4k

Q: Sketch the space curve and find its length over the given interval r(t) = ⟨8 cos t, 8 sin t, t⟩,…

A: Given Finding the derivative with respect to t:

Q: Find T, N, and k for the space curve r(t) = (12 sin t) i + (12 cos t) j+ 16t k. T(t) = () j+ k

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Q: Scalar line integrals Evaluate the following line integral along the curve C.

A: Given integral is Parametric equation of semicircle is

Q: 2. Let y be the curve in R° given by t (t, 2 cos t, 2 sin t) for 0 < t< 47. Compute y? ds

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Q: Q2/ Find T, N, and k for the space curve r(t) = (3 sin t)i +(3cos t)j + 4tk

A: Question: Objective is to calculate the tangent vector T, normal vector N and the curvature κ of…

Q: Find T, N, and k for the space curve r(t) = - 2e cos t)i- (2e sin t) j+ 2k.

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Q: Let C be the curve given by r(t) = (t cost,t sin t,t).(a) Show that C lies on the cone x 2 + y2 = z2

A: Consider the parametric equation of the curve C : rt=tcos t,t sin t, t. To show C lies on cone…

Q: Let F = (2y2, 4x²), and let C be the curve y = x-1 for 1 s x < 4, oriented from left to right.…

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Q: Find T, N, and k for the space curves r(t) = (6 sin 2t)i + (6 cos 2t)j + 5tk

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Q: Find B and t for the space curve below. r(t) = (12 sin 2t)i + (12 cos 2t)j + 7tk 24 cos 2t 24 sin 2t…

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Q: Find T, N, and k for the space curve, where t>0. r(t) = (8 cos t+ 8t sin t)i + (8 sin t- 8t cos t)j+…

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Q: Find the arc length traced out by the endpoint of the vector-valued function 7(t) = t cos t î + t…

A: Solved below.

Q: 7. Given the space curve er(t) = (4 sin t ,4 cos t , 10t), Find T, N, B, and t (torsion)

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Q: Determine the length of the curve (a) F(t) = (cos t, sin*t, 2) on the interval 0<t< %3D (b) F(t) =…

A: Length of curve is given : L = integration (r'(t)dt) from a to b.

Q: Find T, N, and k for the space curves r(t) = (cosh t)i - (sinh t)j + tk

A: Let the given space curve is r(t)=(cosh t)i-(sinh t)j+tk   To determine: The values of the unit…

Q: Find the moving trihedral of the curve R(t) = (e^t)sin(t)i − (e^t)cos(t)j + (e^t)k at any t in the…

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Q: Q4)A) Find B and t for the space curve r(t) = (3 sin t)i + (3 cos t)j + (t). %3D

A: We will find out the required values.

Q: 2. Let L be the line tangent to the space curve represented by r(t) = (t² – In(t))i+(e²-t – t)j+…

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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: Let C be the curve.

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Q: Let f(r, y, 2) = 2ry + 2. Evaluate f(1, y, 2) dz where C is the portion of the helix r(t) = cos ti+…

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Q: Let C be the curve in space given by the following parameterization. r(t) = tỉ + t2j+ 9k, 0 <ts v7.…

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Q: Find T, N, and K for the space curve r(t) = (9 sin t) i + (9 cos t) j+ 12t k. T(t) = (O i+ O i+ (O k

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Q: The value of the line integral | F(r).dr when C F(F) = [-y, -xy]and C is the upper half circle arc…

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Q: Let C be the curve in space given by the following parameterization. r(t) = tỉ + t²j + 9k, 0 < t s…

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Q: Find T, N, and k for the space curve, where t> 0. r(t) = (2 cos t+ 2t sin t)i + (2 sint- 2t cos t)j…

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Q: Let C be the space curve r(t) = (3 sin(t) – 1,4 sin(t) + 1), 0 < t < . The length of C is

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Q: where C is the curve r(t) = (sin(t), cos(t), sin(2t)), 0 ≤ t ≤ 2

A: Given C is the curve                        r(t)=sin(t), cos(t), sin(2t), 0≤t≤2π and…

Q: Let C be the curve parameterized by Y: [0, b] R³, y(s) = (12 cos(s²), 12 sin(s), 9 s²). Find b>0…

A: The given problem is to find the value of b for which the given length of curve of the given…

Q: Consider the space curve r(t) = (cost cos 2t, cost sin 2t, sin t). (a) Find r(t).

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Q: The length of the curve r(t)= [cos(t), sin(t), – 3t]with 0 <t<2n is equal to O A. 2/5 T O B. 2/6 T O…

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Q: 3. Sketch the space curve, r(t) = 8 cos ti +8 sin t j+tk , and find its length over the interval []…

A: Given: r(t)=8cost+8sint+tk    [0,pi/2]

Q: Find the length of the curve. r(t) = cos(6t) i + sin(6t)j + 6In(cos(t)) k, 0stsa/4

A: In this question we have to find the length of the curve.

Q: Let C be the part of the space curve represented by r(t) = (1– e', 2e-, 2t) from t = –1 to t = 1.…

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Q: Find the length of the curve. r(t) = cos(4t) i + sin(4t)j + 4 In(cos(t)) k, 0 <t< t/4

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Q: Find T, N, and k for the space curve r(t) = (16 sin t) i + (16 cos t) j+ 12t k. T(1) = (D i+ (O j+…

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Q: Find T, N, and B for the given space curve. r(1)= (6+t)i + (2 + In (sec t))j-3k, - T/2 <t < T/2 O A.…

A: Find r'(t) by differentiating r(t) with respect to t. r(t)=6+ti+2+lnsec tj-3kr'(t)=i+sec t·tan tsec…

Q: Find an equation of the plane tangent to the following surface at the given point. z = tan (xy);…

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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: 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: Let F(x, y) = -z|. Let C be the curve parametrized by (t, cos(t), sin(t)), where 0 <t<. y Evaluate:…

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Q: 12. Let T be the line tangent to the space curve represented by r(t) = (t³ + e=t )i+ (ln(t) + 2t )j+…

A: First we find the equation of tangent in Cartesian form, and then check whether the tangent…

Q: A) Find B and t for the space curve r(t): = (3 sin t)i + (3 cos t)j + (t)k.

A: As per our guidelines, we are supposed to solve only first question.  Kindly repost other question…

Q: 3. Let C be the part of the space curve represented by r(t) = (t, t/², t² +1), where 1 <t< a. Find…

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Q: 8 Find T, N, and k for the space curve r(t) = Gi+gj, t> 0 8

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Q: Find the length of the following​ three-dimensional curve. r(t)=cos t, sin t, 3t​, for 0≤t≤2

A: Given that rt =cost,sint,3t  for 0≤t≤2 find the length

Q: Consider the space curve r(t) = (-t-1, 4t+3, 2t + 5). a. Find the arc length function for r(t). s(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.