Complex Numbers (Exponential/Rectangular Form)

[Marcin H]

Homework Statement

Homework Equations

Theta = arctan (y/x)

The Attempt at a Solution

Hopefully this is the right section to post in, but I am a bit confused with complex numbers. I am working on the problems above and I just wanted to make sure I am doing each part correctly. I think A and B are correct, but I'm not 100% sure about my theta. That usually trips me up. Are my theta's correct? Did I add pi correctly where it needs it? For C and D do I need to include the angle? In my answers? it does not say what form to do C and D in so I was a bit confused about those. Is there a standard form to use when it's not given in the problem? Also, is there some way to simplify D? OR do I just leave it how I have it?

 

[SammyS]

Homework Statement

View attachment 105323

Homework Equations

Theta = arctan (y/x)

The Attempt at a Solution

Hopefully this is the right section to post in, but I am a bit confused with complex numbers. I am working on the problems above and I just wanted to make sure I am doing each part correctly. I think A and B are correct, but I'm not 100% sure about my theta. That usually trips me up. Are my theta's correct? Did I add pi correctly where it needs it? For C and D do I need to include the angle? In my answers? it does not say what form to do C and D in so I was a bit confused about those. Is there a standard form to use when it's not given in the problem? Also, is there some way to simplify D? OR do I just leave it how I have it?

View attachment 105324
View attachment 105325

It looks like you have done parts (a) & (b) correctly.

Part (c) asks for magnitude, so you're not finished with it.

Part (d):

AB is correct.

I think it's easier to do both AB and A/B using exponential form (also called polar form).

Change answers to rectangular form at the end.​

 

[Marcin H]

Part (c) asks for magnitude, so you're not finished with it.

I thought magnitude only meant the value, without a direction. Is that what I am missing? The direction? If so, how would I find that? Would I have to somehow add the angles I found in part A and B? Angle A + Angle B?

I think it's easier to do both AB and A/B using exponential form (also called polar form).

Change answers to rectangular form at the end.

Oh ok. So since AB is correct I will just leave it. For A/B I can just use the answers from part A and part B right? So when dividing the numbers will just divide like normal but then I can combine the e's to e^(7pi/4 - 5pi/4)j right? Then after that convert to rectangular form? Is that the form you should always go to if the question does not specify?

 

[SammyS]

I thought magnitude only meant the value, without a direction. Is that what I am missing? The direction? If so, how would I find that? Would I have to somehow add the angles I found in part A and B? Angle A + Angle B?
Oh ok. So since AB is correct I will just leave it. For A/B I can just use the answers from part A and part B right? So when dividing the numbers will just divide like normal but then I can combine the e's to e^(7pi/4 - 5pi/4)j right? Then after that convert to rectangular form? Is that the form you should always go to if the question does not specify?

Concerning part (d): The instructions do specify rectangular form.

When you did parts (a) and (b), you found magnitudes by using right triangles. Do the same for part (c).

 

[fresh_42]

I thought magnitude only meant the value, without a direction. Is that what I am missing? The direction? If so, how would I find that? Would I have to somehow add the angles I found in part A and B? Angle A + Angle B?

No. The magnitude is the length. You are asked for $\vert A+B \vert$ and $\vert A-B \vert$.

Oh ok. So since AB is correct I will just leave it. For A/B I can just use the answers from part A and part B right? So when dividing the numbers will just divide like normal but then I can combine the e's to e^(7pi/4 - 5pi/4)j right? Then after that convert to rectangular form? Is that the form you should always go to if the question does not specify?

Not always. It's just easier here. Calculate $A/B$ in polar coordinates and you will see what it is in rectangular form.

SammyS beat me ... but I've done all the calculations, so I couldn't resist ...

 

[thelema418]

I thought magnitude only meant the value, without a direction. Is that what I am missing? The direction? If so, how would I find that? Would I have to somehow add the angles I found in part A and B? Angle A + Angle B?

Oh ok. So since AB is correct I will just leave it. For A/B I can just use the answers from part A and part B right? So when dividing the numbers will just divide like normal but then I can combine the e's to e^(7pi/4 - 5pi/4)j right? Then after that convert to rectangular form? Is that the form you should always go to if the question does not specify?

The magnitude is your $r$ value; other names include the modulus and length. The argument is the $\theta$ value; another name for this is the phase.

For problems involving $A/B$ in rectangular form, you do not need to convert to exponential form to solve the problem. All you have to do is "rationalize" the denominator. With a complex number as the denominator, you multiply by 1 in the form of the complex conjugate of the denominator. The rest is just simplifying the algebra. Example:

$\frac{1}{3+3j} = \frac{1}{3+3j} \times 1 = \frac{1}{3+3j} \times \frac{3-3j}{3-3j} = \frac{3-3j}{(3+3j)(3-3j)} = \frac{3-3j}{18} = \frac{1}{6} - \frac{1}{6}j$