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+ 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: C: r(1) = (1 – cos t)i + sin tj Evaluate F. dr, where F(x, y) = (y + 1)i + (3xy² + 1)j 1- and C, is…

A:

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

A: Given integral is Parametric equation of semicircle is

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.

A:

Q: For the smooth curve r(t), find the parametric equations for the line that is tangent to r at the…

A:

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

A:

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…

A:

Q: If z =t cost, y =tsint, z=t, then x² + y² = t² cos²t+t² sin² t = t² = z², so the curve lies on the…

A: Given x=tcost, y=tsint and z=t and z2=x2+y2 Substitute x=tcost, y=tsint and z=t in z2=x2+y2…

Q: 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

A:

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: Consider the curve C shown in the figure, parameterized by: C: r(t) = (1 - cos(t), 1- sin(t), 1 +…

A:

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…

A:

Q: Consider a particle moving along a path C given by r(t) = v3ti+ = V3ti sin (t)j + cos (t)k Find the…

A:

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

A:

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

A:

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

A:

Q: Find F. dr for F = 4yi + 2x j - 2z k if C is the helix x = cos t, y = sin t, z = t, for 0 <t < 4n.

A:

Q: (a) Evaluate f. (sin x + 3y²)dx + (2x – e-y*)dy, where C is the boundary of the half-disk x² + y? <…

A:

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+…

A:

Q: Find the length of the curve. r(t) : cos(8t) i + sin(8t)j + 8 In(cos(t)) k, 0 stS T/4

A: Solution is given as below:

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

A:

Q: Find T,N and k for the curve: 1- r(t) = (6 sin 2t)i + (6 cos2t) j+5tk.

A: As per our guidelines we are supposed to answer only one question at a time. Kindly repost other…

Q: Consider the curve: r(t) = In (t) i + +(/2)tk Find the length of the curve between the point Po…

A:

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

A:

Q: Let C be the curve parameterized as follows F(t) := (9 cos(t), 9 sin(t), 12t), 0<ts 10 and let f(x,…

A:

Q: Use the Midpoint Rule with N = 10, 20, 30, and 50 to approximate the given curve's length. %3D c(t)…

A: We are given the curve :c(t)=(cos t, esint) The length of the curve: s=∫02π-sin t2+(cos t esin t)2…

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…

A:

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

A:

Q: Given: vector-valued function R(t) whose derivative R'(t) = (4t, 6t, —12t) (c) Determine the…

A:

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: Find the length of the curve. r(t) = cos(4t) i + sin(4t) j + 4ln(cos(t)) k,  0 ≤ t ≤ ?/4

A:

Q: Evaluate F. dr where F = (x – 3y)i + (y – 2r)j and C is the closed curve in the xy plane, x = 2 cos…

A:

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…

A:

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: A curve is written parametrically with x and y defined as functions of t: x=t2 +1, y = 4r° +3. Find…

A: Take the derivative of x and y with respect to t then use the formula for the arc length.

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

A:

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:…

A:

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

A:

Q: Find the length of the curve C defined parametrically by C: r(t) = y(t) = e' sin t, 0<t<n e cost,…

A:

Q: Find the length of the parametric curve x = sin-1t,y = In V1– t² for 0 <t<-

A:

Q: Consider the curve, y with parameter (, and function f in 3 dimensional Euclidean space with…

A:

Q: Let C be the closed curve starting at (0, 1) traveling along the parabola y = (x – 1)? up to the…

A: Using Green's Theorem we solve the above question.

Q: Let C be the part of the curve x? + y² = 1 with y 2 0, oriented counter- clockwise. Evaluate F. dr,…

A:

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.