SOLUTION: 1. Given: Z1= 2-2i , Z2= 3i and Z3= -3+i .Find: (c) Z1*Z3 (d) Z3xZ2 (e) the acute angle between Z1 and Z2

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Question 1194520: 1. Given: Z1= 2-2i , Z2= 3i and Z3= -3+i .Find:
(c) Z1*Z3 (d) Z3xZ2 (e) the acute angle between Z1 and Z2
Found 2 solutions by prakharsingh406, ikleyn:
Answer by prakharsingh406(1)   (Show Source): You can put this solution on YOUR website!
z1=2-2i and z2=3i and z3= -3+1 now
c) z1*z3 = 6i-6i^2 (i^2=-1 as i =
so it will become
6i+6= 6+6i we get
Now I believe you can do d
for e
angle between z1 and z2
z1(2, -2) and z2 (0,3) on a graph
take origin as a reference point now draw on a graph

now y= 3 has 0 slope so angle 0 but 2,-2 has -pie/4 as angle= tan^-1 (-2/2)
now angle between z1 and z2 is z1-z2 is angle between them so -pie/4 -0 =-pie/4
this is the simplest method.
Answer by ikleyn(52788)   (Show Source): You can put this solution on YOUR website!
.

(c)  Z1 * Z2 = (2-2i)*(-3+i) = -6 + 6i + 2i -2i^2 = -6 + 8i -2*(-1) =

             = -6 + 8i + 2 = -4 + 8i.



(e)  The angle (the argument) for Z1 is   = -45°.

     The angle (the argument) for Z2 is   =   90°.


     THEREFORE, the angle between Z1 and Z2 is   -  = .

                Or, in degrees,  the angle between Z1 and Z2 is 90° - (-45°) = 135°.



         This answer is for minimal angle measure.  
     
         The other possible angle "between" these complex numbers is 360° - 135° = 225°,

         but normally, the minimal of the two angles is considered as the answer.

---------------

On complex numbers,  see introductory lessons
    - Complex numbers and arithmetical operations on them
    - Complex plane
    - Addition and subtraction of complex numbers in complex plane
    - Multiplication and division of complex numbers in complex plane
in this site.

Also,  you have this free of charge online textbook on  ALGEBRA-II  in this site
    - ALGEBRA-II - YOUR ONLINE TEXTBOOK.

The referred lessons are the part of this online textbook under the topic  "Complex numbers".


Save the link to this textbook together with its description

Free of charge online textbook in ALGEBRA-II
https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson

into your archive and use when it is needed.



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