Question

What are the components of the vector between the origin and the polar coordinate `(9, (3pi)/4)`?

Answer 1

`-(9sqrt2)/2hati + (9sqrt2)/2hatj`

Explanation:

What we'll do first is find the rectangular form of the given polar coordinate, using the equations

`ul(x = rcostheta`

`ul(y = rsintheta`

where in this case

• `r = 9`

• `theta = (3pi)/4`

So we have

`x = 9cos((3pi)/4) = -(9sqrt2)/2`

`y = 9sin((3pi)/4) = (9sqrt2)/2`

SInce the vector starts at the origin, these values are the `x`- and `y`-components of the vector, so we can write the vector as

`color(blue)(ulbar(|stackrel(" ")(" "< -(9sqrt2)/2, (9sqrt2)/2 >" ")|)`

Or

`color(blue)(ulbar(|stackrel(" ")(" "-(9sqrt2)/2 hati + (9sqrt2)/2 hatj" ")|)color(white)(aa)` (unit vector notation)