#7**+10 **

Hi Melody

As Alan has said $$r\angle \theta$$ is a shorthand notation used mainly by electrical engineers.

It's actually shorthand for the complex number $$r(\cos \theta + \imath \sin \theta).$$

Written out in full, we would have

$$r_{1}\angle \theta_{1}\times r_{2}\angle\theta_{2}\\=r_{1}r_{2}(\cos\theta_{1}+\imath\sin\theta_{1})(\cos\theta_{2}+\imath\sin\theta_{2})\\=r_{1}r_{2}(\cos\theta_{1}\cos\theta_{2}-\sin\theta_{1}\sin\theta_{2}+\imath(\sin\theta_{1}\cos\theta_{2}+\cos\theta_{1}\sin\theta_{2}))\\=r_{1}r_{2}(\cos(\theta_{1}+\theta_{2})+\imath\sin(\theta_{1}+\theta_{2}))\\=r_{1}r_{2}\angle(\theta_{1}+\theta_{2}).$$

Bertie
Jun 15, 2015

#1**+10 **

Multiply the magnitudes together and add the angles:

100∠30*(-10∠-165) = -(100*10)∠(30-165) = -1000∠-135

.

Alan
Jun 14, 2015

#3**+5 **

Electrical engineers commonly use this sort of representation for the magnitude and phase of ac electrical currents; but in general you could use it for any two-dimensional vectors expressed in polar coordinates.

.

Alan
Jun 15, 2015

#4**0 **

Mmm, thanks Alan, I still don't have any concept of what it means.

Does it have a pictorial representation or a situation word problem that I could understand where it would be useful?

Melody
Jun 15, 2015

#5**+5 **

If a vector is given in polar coordinates by (r, θ) then it can also be written as r∠θ (which electrical engineers do a lot!).

Multiplying two together is just a combined stretching (r1*r2) and rotating (θ1 + θ2) transformation.

.

Alan
Jun 15, 2015

#7**+10 **

Best Answer

Hi Melody

As Alan has said $$r\angle \theta$$ is a shorthand notation used mainly by electrical engineers.

It's actually shorthand for the complex number $$r(\cos \theta + \imath \sin \theta).$$

Written out in full, we would have

$$r_{1}\angle \theta_{1}\times r_{2}\angle\theta_{2}\\=r_{1}r_{2}(\cos\theta_{1}+\imath\sin\theta_{1})(\cos\theta_{2}+\imath\sin\theta_{2})\\=r_{1}r_{2}(\cos\theta_{1}\cos\theta_{2}-\sin\theta_{1}\sin\theta_{2}+\imath(\sin\theta_{1}\cos\theta_{2}+\cos\theta_{1}\sin\theta_{2}))\\=r_{1}r_{2}(\cos(\theta_{1}+\theta_{2})+\imath\sin(\theta_{1}+\theta_{2}))\\=r_{1}r_{2}\angle(\theta_{1}+\theta_{2}).$$

Bertie
Jun 15, 2015