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))

To get r:

r = [x

Therefore we may write:

^{2}+ y^{2}]^{1/2}= [4^{2}+ 3^{2}]^{1/2}= [25]^{1/2}= 5Therefore 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)

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)

We look now 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

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

**:**

*Complex Division*
Let's say we
want to divide:

z1 = Ö2(cos(-45) + isin(-45)) = Ö2 cis(-45)

by

z2 = 3.6(cos(56.3) + isin(56.3)) = 3.6 cis(56.3)

In all such cases of complex division we require that the z, r in the denominator not be zero.

Thus:

z1 = Ö2(cos(-45) + isin(-45)) = Ö2 cis(-45)

by

z2 = 3.6(cos(56.3) + isin(56.3)) = 3.6 cis(56.3)

In all such cases of complex division we require that the z, r in the denominator not be zero.

Thus:

(z

_{1}/z_{2}) = (r_{1}cis(q_{1})/ r_{2 }cis(q_{2})) = (r_{1}/ r_{2}) cis (q_{1}– q_{2})
Now: (r1/ r2) = (1.414/ 3.6) = 0.39

And we saw previously:

(q

And we saw previously:

(q

_{1}– q_{2}) = arg(z_{1}) – arg(z_{2}) = (-45) – (56.3) = -101.3
Thus, the basic procedure for
division entails dividing the lengths (r’s) and subtracting the angles (q

_{1}– q_{2}).So:

(z

_{1}/ z

_{2}) = 0.39 (cos (-101.3) + isin(-101.3))

= 0.39((-0.195) + i(-0.98)) = -0.07
+ 0.38i

What about? (1 +
i) ¸ Ö3 – i

The first order of business is to get dividend and divisor each into polar
form, specifically as a (cis) function.

Then (1 + i) = z

Further:

arctan (y1/x1) = arctan (1/1) = arctan (1) so q

What about r1?

r1= [1

_{1}= x1 + iy1, so arg(z_{1}) = arctan (y1/x1)Further:

arctan (y1/x1) = arctan (1/1) = arctan (1) so q

_{1}= 45 degWhat about r1?

r1= [1

^{2}+ 1^{2}]^{1/2}= Ö2 = 1.4
so z

_{1}= 1.4 [cos (45) + isin(45)] = 1.4 cis(45)
Now: z

_{2}= Ö3 – i
So arg(z

_{2}) = arctan(y_{2}/x_{2}) = arctan(-1/ Ö3) so q_{2}
=
(-30 deg)

r

Then subtract angles: [(q

So the end result of the division is:

(z

= 0.707{cos(75) + isin(75)}

Since cos(75) = 0.258 and sin(75) =0.966, we have:

(z

Another very convenient way to express complex numbers is in the exponential form.

Thus, we can write:

Thus, the previous numbers we divided (z

z

z2 = 2[cos(-30) +isin(-30)] = 2 exp(i (-p/6))

1) Express each of the following end results in the form: r exp(iq):

a) (2 + 3i)(1 – 2i)

b) (1 + i) (1- i)

_{1}/r_{2}= Ö2/ 2Then subtract angles: [(q

_{1}– q_{2}) ] = {(45 deg) – (-30 deg)} = 75 degreesSo the end result of the division is:

(z

_{1}/z_{2}) = Ö2/ 2 cis(75) = Ö2/ 2 {cos(75) + isin(75)}= 0.707{cos(75) + isin(75)}

Since cos(75) = 0.258 and sin(75) =0.966, we have:

(z

_{1}/z_{2}) = 0.707[(0.258) + i(0.966)] = 0.183 + 0.683iAnother very convenient way to express complex numbers is in the exponential form.

Thus, we can write:

*cos(**q**) + isin(**q**) = r exp (i**q**)*Thus, the previous numbers we divided (z

_{1 }and z_{2}) may be expressed:z

_{1}= Ö2 [cos (45) + isin(45)] = Ö2 exp (i p/4)z2 = 2[cos(-30) +isin(-30)] = 2 exp(i (-p/6))

*Practice Problems*:1) Express each of the following end results in the form: r exp(iq):

a) (2 + 3i)(1 – 2i)

b) (1 + i) (1- i)

c) (1 + Ö-3)

^{2}2) Plot the results of (b) and (c) on the same Argand diagram and obtain the resultant. Check

_{}

^{}