We left off in the previous instalment obtaining the vector C in complex form, along with its argument.
Now we look at the vectors A and B, which will call henceforth z1 and z2 to be consistent with complex notation. Our eventual goal will be to find the resultant- which will come in the next instalment. In the meantime we will be working toward showing the multiplication and division of two complex forms, call them z1 and z2:
[z1 + z2]
From the diagram (see previous blog):
A= z1 = -2 + 2i
B = z2 = -2 -3i
So:
z1 = x1 + iy1
arg(z1) = arctan(y1/x1) = arctan (-2/2) = arctan(-1)
so (theta_1) = -45 degrees
Now find r1:
r1 =[x1^2 + y1^2]^1/2 = [1^2 + 1^2]^1/2 = [2]^1/2 = 1.414
Therefore:
z1 = 1.414 (cos(-45) + isin(-45)) = 1.414 cis(-45)
We now turn to the vector B which is:
z2 = x2 + iy2= -2 -3i
then: arg(z2) = arctan(y2/x2) = arctan (-3/-2) = arctan (3/2) = 56.3 deg
While:
r2 =[x2^2 + y2^2]^1/2 = [(-2)^2 + (-3)^2]^1/2 = [13]^1/2 = 3.6
Therefore:
z2 = 3.6(cos(56.3) + isin(56.3) = 3.6 cis(56.3)
Now, how do we obtain the complex product: [z1*z2]?
We have that:
[z1*z2] = (z1*z2) cis(arg(z1) – arg(z2))
But:
(z1*z2) = (1.414)x (3.6) = 5.1
And:
arg(z1) – arg(z2) = (-45) – (56.3) = -101.3
so that:
[z1*z2] = 5.1 cis(-101.3) = 5.1 (cos (-101.3) + isin(-101.3))
[z1*z2] = 5.1((-0.195) + i(-0.98))
[z1*z2]= 0.99 + 0.98i
Next Series topic: Complex division and complex resultants
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