Question

What are the components of the vector between the origin and the polar coordinate `(-4, (-7pi)/4)`?

Answer 1

`<-2sqrt2, -2sqrt 2>`.-

Explanation:

In both cartesian `(x, y) and` polar `(r, theta)` forms, the

components of the position vector `OP`, from the origin to the point

P(x,y) are `< x, y > = < r (cos theta, sin theta)>`, repectively

Here, `(r, theta)=(-4, -(7pi)/4)`. and so, the components are

`<-4 cos(-7pi/4), -4 sin(-7pi/4)>`, using cos (-a) = cos a and sin (-a)

=-sin a

`= <-4cos(7pi/4), 4 sin (7pi/4)>`

`= <-4 cos (2pi-pi/4), 4 sin (2pi-pi/4) >`

`= <-4 cos (pi/4)- 4 sin (pi/4) >`,

using `cos (2pi-a)=cos a and sin (2pi-a)=-sina`.

`= <-4/sqrt2, -4/sqrt2 >`

`= <-2sqrt2, -2sqrt 2>`.-