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

1 Answer
Jun 20, 2016

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!)