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A simple angle-vector problem!

  1. Feb 28, 2012 #1
    1. The problem statement, all variables and given/known data

    Consider a vector z defined by the equation z = z1z2, where z1 = a + jb, z2 = c + jd

    Show that the angle between z and the x axis is the sum of the angles made by z1 and z2 separately.

    (EDIT): PLEASE LOOK AT MY BELOW POST!!
     
    Last edited: Feb 28, 2012
  2. jcsd
  3. Feb 28, 2012 #2

    tiny-tim

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    Hi JJHK! :wink:

    show us what you've tried, and where you're stuck, and then we'll know how to help! :smile:
     
  4. Feb 28, 2012 #3
    Okay, well here's actually the first part of the problem. I also forgot to define what j is. It is an instruction to perform a counterclockwise rotation of 90°.

    Now here's the first part:

    Consider a vector z defined by the equation z = z1z2, where z1 = a + jb, z2 = c + jd

    (a) Show that the length of z is the product of the lengths of z1 and z2

    I first found the length of z1 and z2, L(z1) and L(z2):

    L(z1) = √(a2+b2)

    L(z2) = √(c2+d2)

    Now I'm going to find the length of z, L(z):

    L(z) = √((ac-bd)2+(ad+bc)2)
    = √(a2c2+a2d2+b2c2+b2d2)

    I'm going to show that the above solution is equal to L(z1)L(z2)

    L(z1)L(z2) = √(a2+b2)√(c2+d2)
    = √(a2c2+a2d2+b2c2+b2d2)

    Therefore, L(z) = L(z1)L(z2)

    Now part 2:

    (b)Show that the angle between z and the x axis is the sum of the angles made by z1 and z2 separately.

    I first found the angles of z1 and z2, θ(z1) and θ(z2):

    θ(z1) = arctan(b/a) and θ(z2) = arctan(d/c)

    and the angle for z is:

    θ(z) = arctan((ad+bc)/(ac-bd))

    Now i'm stuck, how do I equate θ(z) = θ(z1) + θ(z2) ??

    THanks for the help!
     
  5. Feb 28, 2012 #4

    tiny-tim

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    so far so good! :smile:

    hint: if tan-1A = tan-1B + tan-1C,

    then A = tan(tan-1B + tan-1C) …

    and what's the formula for tan of a sum? :wink:
     
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