arctan (y/x) = arctan(3/4) = 36.8 deg

Thus, Θ = 36.8 degrees is the argument

Now, any complex number (x + iy) may be written in polar form:

x + iy = r(cos (Θ) + isin(Θ))

And to get r:

r = [x

^{2}+ y

^{2}]

^{1/2}= [4

^{2}+ 3

^{2}]

^{1/2}= [25]

^{1/2}= 5

Therefore we may write: (x + iy) = 5(cos (36.8) + isin(36.8))

Note there is also the abbreviated function (based on the combo of sine and cosine):

cis (Θ) = cos (Θ) + isin(Θ)

so we can finally write:

C = r cis(Θ) = 5 cis (36.8)

Now we look at the vectors A and B, which we’ll henceforth call z1 and z2 to be consistent with complex notation. Our eventual goal will be to find the resultant, which will come in the next installment. In the meantime we will be working toward showing the multiplication and division of two complex forms, call them z1 and z2:

e.g. [z1 + z2]

From the diagram:

A= z1 = -2 + 2i

B = z2 = -2 -3i

So: z1 = x1 + iy1

And arg(z1) = arctan(y1/x1) = arctan (-2/2) = arctan(-1)

So (Θ1) = -45 degrees = -π /4

Now find r1:

r

_{1}=[x

_{1}

^{2}+ y

_{1}

^{2}]

^{1/2}= [1 + 1]

^{1/2}= Ö2 Therefore:

z1 = Ö2 (cos(-45) + isin(-45)) = Ö2 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:

r

_{2}= [x

_{2}

^{2}+ y

_{2}

^{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) = Ö2 (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

To get the resultant: z1 + z2 = z3:

A + B = z1 + z2 =[ (-2 + 2i) + (-2 – 3i)] = -4 –i

In any case:

x3 + iy3 = - 4 – i

Problems for the Math Maven:

1) Based on the resultant for (x3, y3) obtain arg(z

_{3}) and thence the polar form for the resultant. (Hint: Remember arg(z

_{3}) = Θ )

2) Now combine this resultant z

_{3 }with that from C (Fig. 1) which we call z

_{4}, to obtain z

_{5}

Thence, find arg(z

_{5}) and write in polar form.

3) Obtain the complex product for [z3• z4]

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