Consider the following space curve r(t) = ti+ 2tj + In(t)k a) Calculate the velocity v(t). b) Calculate the speed |v(t)| and simplify by completing the square. Hence find the distance travelled between times t = 1 and t = 2. |3D c) Find the time points when v(t) is parallel to the plane -x + y = 0.

Q: By using Green's theorem evaluate £ (Ary – y") dr + (** + 3ry) dy where C is a closed curve starting…

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Q: Let C be the boundary of the portion of the plane x+y+z=2 that lies in the first octant, oriented…

A: It is a 3D and we have to draw in a sheet which is 2D. I is a tough task but I will try to draw it…

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Q: Find an equation of the tangent plane to the surface S : za? – xy – yz = 18 at the point (0, –2, 3).…

A: We know that an equation of the tangent plane to the surface Fx, y, z at the point x0, y0, z0 is…

Q: Given the surface z = In(-2x² + 5y² – xy). Find the directional derivative of the function f at…

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Q: Let C be the curve in the plane with parametrization r(t) = cos ti+ sin' tj for t e [0, 47]. Compute…

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Q: Evaluate F · dī where C F = , and C is the circle of radius 3 centered at the origin parametrized…

A: To find  ∫CF→.dr→ where c is circle of radius 3 centered  at origin parametrized in counterclockwise…

Q: Find the curvature of the space curve. r (1) = -8i + (1 + 4) j + (8 + In(cos t)) k %3D OK = sec t…

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Q: Let D be the half-disc in the xy-plane given by x +y <9. Calculate e-(x2+v)dA

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Q: Sketch the space curve r(t) = ⟨2 sin t, 5t, 2 cos t⟩ and find its length over the given interval [0,…

A: The given equation of space curve is  r(t) = ⟨2 sin t, 5t, 2 cos t⟩.

Q: In which part of the (x, y) plane the following equation is elliptic? Uzr + 4ury + (x² + 4y*) uyy =…

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Q: Sketch the space curve r(t) = i + t2j + t3k and find its length over the given interval [0, 2] .

A: If r=x(t)i+y(t)j+z(t)k then length of r over [a, b] is L=∫ab[x'(t)]2+[y'(t)]2+[z'(t)]2dt

Q: Consider the curve C parametrized byx = (t^2+1)/(t^2-1) and y = (2t)/(t^2-1) for all t in (−∞, −1 )…

A: Given:

Q: Find the curvature of the plane curve with equation y (x, f (x)) on the curve, by using the…

A: We will find out the required value of curvature .

Q: Find a parametrization of the line in which the planes x+y +z= -2 and y +z= 6 intersect.

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Q: Let C be the parametric curve defined by Sx(t) = t2 – t – 2, |v(t) = t* – º +t. Determine if C is…

A: Substitute x=-2 in the given equation xt=t2-t-2 and simplify to calculate the value of t.…

Q: Let C be the space curve defined by Ř(t) = (2cos t,2 sin t, t). O Find the moving trihedral of C at…

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Q: Sketch the space curve r(t) = −ti + 4tj + 3tk and find its length over the given interval [0, 1] .

A: Given space curve is rt=-ti+4tj+3tk. Sketch the above space curve as shown below.

Q: Find the equation for the tangent plane to the surface -3x+ 9y -7z = -26 at the point (1,-1, Lütfen…

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Q: 2r find the moment with respect to the wavic of the area hounded by the parabola v - Av-172 and v

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Q: Find a parametrization for the line segment joining points (0, 2) and (4, 0) using the angle 0 in…

A: Modify the given figure as shown in Figure 1.

Q: Evaluate F. dī where F = , and C' is the circle of - counterclockwise direction.

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Q: 6. Given the space curve r(t) = (t, 2et, e2t), find the tangential and normal components of…

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Q: Sketch the space curve r(t) = a cos ti + a sin tj + btk and find its length over the given interval…

A: Given : Space curve equation rt = acosti^ + asintj^ + btk^  ,   and  t ∈ 0,2π  ,a and b are contants…

Q: Compute the length of the curve given parametrically by x=1/3t^3 and y=1/2t^2 for 0 less or equal to…

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Q: Calculate Se(5x + 3y) i + (4x + 3y) j) · dĩ where C is the circular path with center (a, b) and…

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Q: Find a parametrization for the line through the point (a, b) having slope m.

A: Given that slope = m , this can be written as &lt;1,m&gt; vector. And point= (a,b)

Q: Sketch the plane curve. r(t) = 4ti - tj, [0, 4] y y 4 4 2 10 15 10 15 o-4 o-4 y y 4 X 5 10 15 10 15…

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Q: Give the length of a curve given parametrically by x = x(t), y = y(t), and z = z(t) for a ≤ t ≤ b in…

A: given parametrically by x = x(t), y = y(t), and z = z(t) for a ≤ t ≤ b in the vector form.

Q: Find the length of parametrized curve given by x(t)=−6t^2−6t,y(t)=−4t^3−6t^2 where t goes from 0…

A: Given information:  The parametric equations are x(t)=−6t2−6t,y(t)=−4t3−6t2 over the interval .0≤t≤1…

Q: Use the Green's theorem to calulate So (3x²y – xy? – 3.x²y²) dx + (x³ – 2.x³y+ x²y) dy, where C is…

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Q: Find a vector parametrization of the circle of radius 5 in the xy-plane, centered at the origin,…

A: Solution: Given, Circle having center at (0, 0) and radius is 5 and

Q: . Show that the tangent plane to the surface 2yyo cz at the point P(xo,Y0, zo) is given by the 2rro…

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Q: Determine the torsion at t = 0 of a curve a : [−1, 1] → R3, a(t) = (2t, t^2, t^3/3). Moreover,

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Q: : The curve C is intersecting by 12r² +9y? = 36 cylinder and - V3r plane from the point (0, 2,0) to…

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Q: Find the equation for the tangent plane to the surface z= In (7x + 10y + 1) at the point (0,0,0).

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Q: Sketch the space curve r(t) = ⟨4t, −cos t, sin t⟩ and find its length over the given interval [0,…

A: Consider the given space curve, rt=4t,-cost,sint use the graphing utility to graph the curve,

Q: Give an example of a parametrisation of the unit circle, C = {(x,y) E R² |x²+y² = 1}, that generates…

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Q: Find a plane curve a(s) endowed with a unit-speed prametrization by SE (-0, 0) such that its signed…

A: The curvature of the curve αs is given by: αs=∫s0scos(θ(u))du, ∫s0ssin(θ(u))du-----(1) in which…

Q: Find the curvature K for the space curve r(t) = (/3sint)i + (sint)j+ (2cost)k.

A: Let, γ(t) be a space curve. The curvature of the space curve is given by, κ=γ'(t)×γ"(t)γ'(t)3.

Q: Show that for a plane curve the torsion T = 0.

A: We have to shown that, For a plane curve the torsion T=0

Q: a. Find the equation of osculating plane to Z1 at t=0

A: A(0) = &lt; -2, 5, 4&gt;     ---(1)  B(t) = &lt; 3/5 cos t , 3/5 sin t , 4/5 &gt;    =&gt; B(0)  =…

Q: Find the curvature K of the plane curve y 3 at x = 3. 2х + K =

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Q: Find the curvature of the space curve. r(t) = (6t sin t+ 6 cos t)i + 6j + (6t cos t-6 sin t)k 1 O A.…

A: Given space curve

Q: Let C be the curve in the plane with parametrization r(t) = cos ti+ sin tj for t e [0, 4T]. Compute…

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Q: Consider the plane curve parametrized by F(1) -i+ (In(com(1)J, Find curvature s(4).

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Q: Find a vector parametrization of the circle of radius 5 in the xy-plane, centered at the origin,…

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Q: Find a parametrization for the line segment joining points (0, 2) and (4, 0) using the angle 0 in…

A: Given: The line segment joining (0, 2) and (4, 0). Do the construction as in below figure

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.