Question

What are the components of the vector between the origin and the polar coordinate `(-6, (17pi)/12)`?

Answer 1

The `x` component is `1.55`
The `y` component is `5.80`

Explanation:

The components of a vector are the amount the vector projects (i.e. points) in the `x` direction (this is the `x` component or horizontal component) and `y` direction (the `y` component or vertical component).

If the co-ordinates you'd been given were in Cartesian co-ordinates, rather than polar co-ordinates, you'd be able to read the components of the vector between the origin and the point specified straight from the co-ordinates, as they'd have the form `(x,y)`.

Therefore, simply convert into Cartesian co-ordinates and read off the `x` and `y` components. The equations that transform from polar to Cartesian co-ordinates are:
`x = r cos(\theta)` and
`y = r sin (\theta)`

The form of the polar co-ordinate notation you've been given is `(r, \theta) = (-6, \frac{17\pi}{12})`. So substitute `r = -6` and `\theta = \frac{17\pi}{12}` into the equations for `x` and `y`.

`x = -6 cos (\frac{17\pi}{12})`
`x = (-6) (-0.25882)`
`x = 1.5529`
`x \approx 1.55`

`y = -6 sin(\frac{17\pi}{12})`
`y = (-6)(-0.96593)`
`y = 5.7956`
`y \approx 5.80`

The co-ordinate of the point is therefore `(1.55,5.80)`.

The other end of the vector is at the origin, and so has co-ordinate `(0,0)`. The distance it covers in the `x` direction is therefore `1.55-0 = 1.55` and the distance it covers in the `y` direction is `5.80-0 = 5.80`.

The `x` component is `1.55` and the `y` component is `5.80`.

I highly recommend you have a look at this page on finding components of vectors. It works with polar and Cartesian co-ordinates, like you have done here, and has some diagrams that will make the process make sense. (There are lots of worked examples similar to this as well!)