Curve: r(t) = b cos ti+ b sin tj + V1 - 6² tk t= 2n [c) t = 0 9. One turn of a helix

Q: For long cirular cfinder of Re two halves maintained at temprature U1 and Vi respectively, Show that…

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Q: 3. Compute : exp (sin (e).

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Q: Show that the work of F = (2³-y³)i + (2³-y³)j on c, the unit circle centered at the origin with…

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Q: c.) S (4t²-t²+16t)dt/(t²+4) d.) S V1-cos*o de

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Q: $ Evaluate clockwise. positiver sin z 22 +1 dz where y is circle C(0, 2) traversed once and counter-

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Q: find the speed at the given value of t. r(t) = (sin 3t, cos 4t, cos 5t), t =

A: Given: rt=sin3t, cos4t, cos5t To find the speed at the given value of t=π2. Solution: Speed is the…

Q: Compute sin z dz , where a is the piece of the parabola with equation y= a?, which lies between the…

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Q: y (t) = ci + Czet+ c3 cos t + c4 sin t + cst cost + cst sin t.

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Q: Using "Translation Theorems" of Section 4.3, perform the indicated operations: 1 I {t°e 7*} I{e *…

A: Note "As per our company guidelines we are supposed to answer ?️only first 3 sub-parts. Kindly…

Q: Find the length of the hypocycloid defined by the equations x = a cos° (0), y = a sin° (0), 0 < 0 <…

A: Here we have to find the length of the hypocycloid defined by the equation x=acos3(θ) , y=asin3(θ)

Q: The curves r₁(t) = (5t, t², t3) and r₂(t) = (sin(t), sin(5t), 4t) intersect at the origin. Find…

A: see my solution below

Q: The curves T1 (t) = (3t, t°, – 4t°) and r2(t) = (sin(5t), sin( – 2t), t) intersect at the origin.…

A: We have to find the direction vector first.

Q: Find the arclength L(C) of the helix C parametrized by C: r(t) = (2 cos t, -3t, 2 sin t) for 0 <t <…

A: The arc length of the parametric curve is given by: ∫ab(x'(t))2+(y'(t))2+(z'(t))2dt...(1); a≤t≤b…

Q: Find the Cartesian equation of the curve: [2r/r² = 1+ cos a]

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Q: A circular wheel with radius 1 ft andcenter C rolls to the right along the x-axis at a half-turn per…

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Q: c) Determine the Cartesian form of the following equations. i) x = sin{(R + 1)t}, y = cos{2(R +…

A: Given: x=sinR+1t, y=cos2R+1t We know that cos2x=1-2sin2x…

Q: Find the unit to the - tangent r = (a cos t, a sin t. h t) − ∞ 0 and equation of tangent complete…

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Q: The curves F₁ (t) = (2 sin(t), 4 cos(t), e7t) and 72(t) = (3 ln(t), 4√t, t³) intersect at the point…

A: The angle between two curves is the angle between their tangents at the point of intersection .

Q: A particle travels along the path of a helix with the equation r(t) = cos(t)i+ sin(t)j+tk. See the…

A: Consider a vector r(t), which is differentiable. Then a unit tangent vector along the curve r(t) is…

Q: Find the length of the curve. r(t) = (7t, 3 cos t, 3 sin t), -2 ≤ t ≤ 2

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Q: Find the equation of the plane that is perpendicular to the curve  C(t) = at t = π

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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.