Complex Numbers - Argand Plane
Suppose that we have two complex numbers
z
1
=
3
−
2
i
and
z
2
=
4
−
3
i
.
In what quadrant is this complex number z 1 z 2 located?
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2 solutions
If we take the product
z 1 z 2 = = = = = = ( 3 − 2 i ) ( 4 − 3 i ) ( 3 ) ( 4 ) − 3 ( 3 i ) − 4 ( 2 i ) + ( 2 i ) ( 3 i ) 1 2 − 9 i − 8 i + 6 i 2 1 2 − 1 7 i + 6 ( − 1 ) 1 2 − 1 7 i − 6 6 − 1 7 i .
In the Argand plane, this is the equivalent of the point ( 6 , − 1 7 ) , which is in quadrant IV.
The result as product of two complex numbers given is
z 3 = ( 3 − 2 i ) ( 4 − 3 i ) = 1 2 − 1 7 i + 6 i 2 = 6 − 1 7 i
The new complex number we get is z 3 = 6 − 1 7 i
R e ( 6 ) , I m ( − 1 7 )
Hence, in the Argand plane, it will be in the IVth quadrant