Question

What are the components of the vector between the origin and the polar coordinate `(6, (2pi)/3)`?

Answer 1

`((-3),(3sqrt3))`

Explanation:

To convert from `color(blue)"Polar to Cartesian coordinates"`

That is `(r,theta)to(x,y)`

`color(orange)"Reminder"`

`color(red)(|bar(ul(color(white)(a/a)color(black)(x=rcostheta,y=rsintheta)color(white)(a/a)|)))`

here r = 6 and `theta=(2pi)/3`
`color(blue)"-------------------------------------"`
`rArrx=6cos((2pi)/3)`

`color(orange)"Reminder"`

`color(red)(|bar(ul(color(white)(a/a)color(black)(cos((2pi)/3)=-cos(pi-(2pi)/3)=-cos(pi/3))color(white)(a/a)|)))`

`rArrx=-6cos(pi/3)=-6xx1/2=-3`
`color(blue)"--------------------------------------------------------------"`

and y `=6sin((2pi)/3)`

`color(orange)"Reminder"`

`color(red)(|bar(ul(color(white)(a/a)color(black)(sin((2pi)/3)=sin(pi-(2pi)/3)=sin(pi/3))color(white)(a/a)|))`

`rArry=6sin(pi/3)=6xxsqrt3/2=3sqrt3`
`color(blue)"----------------------------------------------------"`

Thus the components of the vector`=((-3),(3sqrt3))`