Answer:
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
The vector parametric equation for the circle C that starts at the point (3,0) and travels around the circle once counterclockwise for 0≤t≤2π is:
[tex]r(t) = <3cost, 3sint>[/tex]
We have
[tex]f(x, y) = <2y, -sin(y)>[/tex]
As C is a circle centered at the origin (0, 0), with radius r = 3,
So, equation is:
[tex](x - 0)^2 + (y - 0)^2 = r^2[/tex]
[tex]x^2+ y^2 = 9[/tex]
As, cos²β + sin²β = 1
Take ,
x = 3cosβ
y = 3sinβ
Then
[tex]x^2 + y^2 = 3^2cos^2\beta + 3^2sin^2\beta\\= 9(cos^2\beta + sin^2\beta) \\= 9\\x^2 + y^2 = 9[/tex]
The vector parametric equation r(t) is therefore given as
[tex]r(t) = <x(t), y(t)>[/tex]
[tex]= <3cost, 3sint>[/tex]
For 0 ≤ t ≤ 2π
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