Compute the unit binormal vector and torsion of the curve. r(t) = (11t, 5 cos t, 5 sin t) (5, 11 sin t, - 11 cos t) V146 11 O A. B(t) = 146 (0, 11 cos t, - 11 sin t) V146 1 O B. B(t) = 146 (0, - 11 cos t, - 11 sin t) V146 11 OC. B(t) = 146 (5, - 11 sin t, 11 cos t) V 146 11 O D. B(t) = 146

Q: 2. Find T, N the unit bi-normal vector B, and k at the given t value. (2b) r(t) = ( cost, sint, In…

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Q: Given the vector function - r(t) = (t²- 4t)i + sin(2t − 4)3 + ( + ) Find the derivative of r(t). a)…

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Q: (9.) r(t) = (1, cos t, 2 sin t)

A: As per your query the solution of question no 9 is given the next step.

Q: A. B. Let the position vecor be R(t) = (-t, 4 cos (2t), 4 sin(2t)). Compute the principal unit…

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Q: 5. Let 0 be the angle between vectors ū = (1,3) and i = (5, 2). What is cos 0 and what is sin 0?

A: Angle between two vectors is given by,

Q: (c) Find dy/dx at t = T/3. (d) Include in sketch for (a) the tangent line to the curve at t = T/3…

A: We have to find the equation of tangent line.

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: Ifr(t) = (sin t , cos t, 29- t), then determine a vector that is perpendicular to both unit tangent…

A: Given: r(t) = sin t, cos t, 29-t To determine: A vector that is perpendicular to both unit tangent…

Q: Let r(t) = (sin 3t +2t)i + (teʻ)j+(t° + 2t)k. Find the unit tangent vector at the point where t = 0.…

A: The unit tangent vector can be calculated using the differentiation. First obtain the derivative of…

Q: 2. For this entire page, let r(t) = (4 sin(t), 3t, 4 cos(t)) (a) Circle the correct graph for the…

A: The objective of this question is to understand the properties of the vector function r(t) = (4…

Q: The monkey saddle z = x³ - 3xy² can be intersected with the cylinder x² + y² = 1 to produce the…

A: Given monkey saddle is z=x3-3xy2 and the cylinder is x2+y2=1. The given curve is the intersection of…

Q: Find the unit tangent vector of the given curve. r(t) = (7t cost-7 sin t)j + (7t sin t + 7 cost) k…

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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: 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: 5. Find the parametric equations of the line tangent to the curve C defined by the vector equation…

A: To find the parametric equation of the given tangent to the curve. The vector equation of the curve…

Q: Let r(t) = (-4t¹ - 4, 4e-2t, sin(-3t)) = Find the unit tangent vector T(t) at the point t T (0) = 0.…

A: Given; r(t)=<-4t4-4, 4e-2t, sin(-3t)>.

Q: Which of the following vector-valued functions represent the same graph? (Select all that apply.)…

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Q: r'(to) TIr(to)|| A vector-valued function and its graph are given. The graph also shows the unit…

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Q: Find the domain of the vector-valued function t R(t) = tan-¹t² + —_— 3 + sin t² k. 1-1²

A: Given the vector valued function R(t)=tan-1t i^+t1-t2j^+sin t2k^. We have to find the domain of…

Q: Q4. A point is moving along a curve that is given as: R(1) = {cos 31 )i + (sin 31) j + (31)k. Find:…

A: Given position vector is R→t = cos3t i +sin3t j +3t k Function fx, y, z =x y z

Q: Identify the direction from graph of functions given below: vector i) r = i(2 cos 21)+ j(2sin 2r) ;…

A: Given, r→=i^(2cos2t)+j^(2sin2t) Identify the direction vector, From the given position vector,…

Q: = 27. If f(x, y) = −3x² + 4y², find the value of the directional derivative at the point (1, -1) in…

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Q: Choose the direction vector for which the function f defined by f(x,v) = sin(7x)cos(13y) has the…

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Q: Let 7(t) = (3 cos(t), 3 sin(t), 3t) and P = (3, 0,6). Consider the curve c parametrized by F(t).…

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Q: 11) Draw the curve F(t) = ß cos(1) 7+ y sin(t)] + tk. Label some key reference points, and indicate…

A: The given parametric curve, r¯t=7 costi⇀+3sintj⇀+tk⇀. To Sketch: The curve and label its points and…

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: