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Size:  346.1 KBCan someone help me with question 8?I got part one done but i couldn't show argument of u is -1/2pi. I couldn't get part 2 and 3. It would be great if someone drew out an illustration too to help me!
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    What's your simplification of u?
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    (Original post by old_engineer)
    What's your simplification of u?
    1/5-3i
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    (Original post by Snowie9)
    1/5-3i
    That doesn't look right. You need to be multiplying both the top and bottom of u by (1 - 2i). What do you get when you do that?
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    (Original post by Snowie9)
    Name:  IMG_8696.JPG
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Size:  346.1 KBCan someone help me with question 8?I got part one done but i couldn't show argument of u is -1/2pi. I couldn't get part 2 and 3. It would be great if someone drew out an illustration too to help me!
    1. For part i) note that

    a) |u| = \dfrac{|6-3i|}{|1+2i|} so just work out top and bottom and divide

    b) \arg u = \arg(6-3i) - \arg(1+2i).

    Now note that the triangles formed by the re and im components of the complex numbers are similar, so they have angles in common. Write \alpha, \beta for the angles made with the re axis and show graphically or algebraically that \alpha - \beta = -\frac{\pi}{2}

    2. For part ii) note that \arg z = \frac{\pi}{4} is true for all complex numbers which make an angle of 45 degrees with the re axis. Remember that z is drawn as an arrow from the origin - so where do the complex numbers lie that satisfy that equation?

    Note also that z-u=z+(-u) shifts z in the +ve im direction by |u|. (It's the +ve im direction since \arg u = -\pi/2  \Rightarrow \arg -u =\pi/2). This is a translation in the complex plane.

    So \arg(z-u) = \frac{\pi}{4} asks you to find those complex numbers which are shifted by that transformation onto the line you found earlier. The answer to the question can then be solved graphically very easily.

    3. For part iii) note that |z-a| is the distance from a to z, so |z-a|=1 is true for all complex numbers z that are a distance 1 from a. What shape are you then working with?
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    (Original post by atsruser)
    1. For part i) note that

    a) |u| = \dfrac{|6-3i|}{|1+2i|} so just work out top and bottom and divide

    b) \arg u = \arg(6-3i) - \arg(1+2i).

    Now note that the triangles formed by the re and im components of the complex numbers are similar, so they have angles in common. Write \alpha, \beta for the angles made with the re axis and show graphically or algebraically that \alpha - \beta = -\frac{\pi}{2}

    2. For part ii) note that \arg z = \frac{\pi}{4} is true for all complex numbers which make an angle of 45 degrees with the re axis. Remember that z is drawn as an arrow from the origin - so where do the complex numbers lie that satisfy that equation?

    Note also that z-u=z+(-u) shifts z in the +ve im direction by |u|. (It's the +ve im direction since \arg u = -\pi/2  \Rightarrow \arg -u =\pi/2). This is a translation in the complex plane.

    So \arg(z-u) = \frac{\pi}{4} asks you to find those complex numbers which are shifted by that transformation onto the line you found earlier. The answer to the question can then be solved graphically very easily.

    3. For part iii) note that |z-a| is the distance from a to z, so |z-a|=1 is true for all complex numbers z that are a distance 1 from a. What shape are you then working with?
    Thanks for trying to explain to me. For part (i) i got my u= -1/5 - 3i and i got the modulus |u| correct but the arg(u) i keep getting +86.1. I don't know how.
    I really appreciate the effort in trying to explain for me but could u write out the workings so i know how to solve part 2 and 3? I'm really weak in complex number
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    (Original post by Snowie9)
    Thanks for trying to explain to me. For part (i) i got my u= -1/5 - 3i and i got the modulus |u| correct but the arg(u) i keep getting +86.1. I don't know how.
    But u doesn't equal -1/5-3i, so you have an error somewhere. Can you put up your working please?

    I really appreciate the effort in trying to explain for me but could u write out the workings so i know how to solve part 2 and 3? I'm really weak in complex number
    The rules of this forum forbid full solutions, and it wouldn't help you much anyway if I did your homework, as I won't be there in the exam.
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    (Original post by Snowie9)
    Thanks for trying to explain to me. For part (i) i got my u= -1/5 - 3i and i got the modulus |u| correct but the arg(u) i keep getting +86.1. I don't know how.
    I really appreciate the effort in trying to explain for me but could u write out the workings so i know how to solve part 2 and 3? I'm really weak in complex number
    Your u= -1/5 - 3i is still wrong. It just happens to give quite a close approximation to the correct mod(u). I think it would be best if you posted your working on the simplification of u, as you'll never make progress with the rest of the question unless that part is right.
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    (Original post by atsruser)
    But u doesn't equal -1/5-3i, so you have an error somewhere. Can you put up your working please?


    The rules of this forum forbid full solutions, and it wouldn't help you much anyway if I did your homework, as I won't be there in the exam.
    Name:  IMG_8721.jpg
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Size:  497.2 KBThis is my attempt on part (i). Yeah you're right..because normally i would understand how to do after looking at solutions but i guess it wouldn't help during exams..
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    (Original post by old_engineer)
    Your u= -1/5 - 3i is still wrong. It just happens to give quite a close approximation to the correct mod(u). I think it would be best if you posted your working on the simplification of u, as you'll never make progress with the rest of the question unless that part is right.
    I posted my working on part(i).
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    (Original post by Snowie9)
    I posted my working on part(i).
    -15i / 5 = -3i

    (-15i has no real component, and the same applies if you divide it by a real constant).
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    (Original post by Snowie9)
    Name:  IMG_8721.jpg
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Size:  497.2 KBThis is my attempt on part (i). Yeah you're right..because normally i would understand how to do after looking at solutions but i guess it wouldn't help during exams..
    \frac{-15i} 5 = \frac{-15} 5 i = -3i

    You have seemingly treated the numerator as 1-15i.

    Also, how did you get 3 for your modulus? (it's correct, but it shouldn't have been. I'd make sure you're working out moduli correctly)
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    (Original post by _gcx)
    \frac{-15i} 5 = \frac{-15} 5 i = -3i

    You have seemingly treated the numerator as 1-15i.

    Also, how did you get 3 for your modulus? (it's correct, but it shouldn't have been. I'd make sure you're working out moduli correctly)
    I guess my mistake somehow got the modulus right. Careless mistake. But how do you carry on to part two?
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    (Original post by old_engineer)
    -15i / 5 = -3i

    (-15i has no real component, and the same applies if you divide it by a real constant).
    My careless mistake
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    (Original post by Snowie9)
    I guess my mistake somehow got the modulus right. Careless mistake. But how do you carry on to part two?
    What does the locus of \arg z = \frac \pi 4 look like? (consider that \tan \frac \pi 4 = 1) If we apply the transformation to z \mapsto z+3i, how does that change the locus? How can we then find the minimum distance between the origin and the line? (writing in Cartesian form may help here)
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    (Original post by _gcx)
    What does the locus of \arg z = \frac \pi 4 look like? (consider that \tan \frac \pi 4 = 1) If we apply the transformation to z \mapsto z+3i, how does that change the locus? How can we then find the minimum distance between the origin and the line? (writing in Cartesian form may help here)
    This is how i can understand from your explanation. I'm sorry if i couldn't get it fully :/
 
 
 
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