(1 point) Suppose F⃗ (x, y)=⟨2y,−sin(y)⟩ and C is the circle of radius 3 centered at the origin oriented counterclockwise. (a) Find a vector parametric equation r⃗ (t) for the circle C that starts at the point (3,0) and travels around the circle once counterclockwise for 0≤t≤2π.

The required vector parametric equation is given as:

r(t) = <3cost, 3sint>

For 0 ≤ t ≤ 2π

Step-by-step explanation:

Given that

f(x, y) = <2y, -sin(y)>

Since C is a cirlce centered at the origin (0, 0), with radius r = 3, it takes the form

(x - 0)² + (y - 0)² = r²

Which is

x² + y² = 9

Because

cos²β + sin²β = 1

and we want to find a vector parametric equations r(t) for the circle C that starts at the point (3, 0), we can write

x = 3cosβ

y = 3sinβ

So that

x² + y² = 3²cos²β + 3²sin²β

= 9(cos²β + sin²β) = 9

That is

x² + y² = 9

The vector parametric equation r(t) is therefore given as

r(t) = <x(t), y(t)>

= <3cost, 3sint>

For 0 ≤ t ≤ 2π

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