Calculate z. w for the given complex numbers z and w. Verify that the length, resp. angle, of z. w is the product of lengths, resp. sum of angles, of z and w. (a) z=1-j; w=2j (b) z=3+j; w=-1-2j

a. Given that z=1-j and w=2j

Then, z. w = 1*0+-1*2 = 0-2 = -2

Then, z. w=-2. Answer

Also, z. w = |z||w|Cosθ

|z| = sqrt (1^2 + (-1) ^2) = √2

|w| = sqrt (0^2+2^2) = √4=2

Therefore, Cosθ = z. w/|z||w|

Cosθ=-2 / (2*√2) = 1/√2

θ=arcCos (1/√2) = 45

For z=1-j

The angle is arctan (y/x)

Arctan (-1/1) = - 45

For w=2j

The angle is arctan (y/x)

Arctan (2/0) = 90.

Sum of angle is - 45+90 = 45

Therefore, the sum of the individual angles z and w is equal to the angle of z. w

|z||w| = 2 * √2=2.83

This product of the individual length of z and w is not equal to the length z. w, this is because the angle between them is 45 assuming the angle is 0 then it will be equal.

Or this can also happen when the two lines are parallel.

b. Given that z=3+j and w=-1-2j

Then, z. w = 3*-1+1*-2 = - 3-2 = -5

Then, z. w=-5. Answer

Also, z. w = |z||w|Cosθ

|z| = sqrt (3^2+1^2) = √10

|w| = sqrt ((-1) ^2 + (-2) ^2) = √5=

Therefore, Cosθ = z. w/|z||w|

Cosθ=-5/√50

θ=arcCos (-5/√50) = 135 or - 225

For z=3+j

The angle is arctan (y/x)

Arctan (1/3) = 18.43

For w=-1-2j

The angle is arctan (y/x)

Arctan (-2/-1) = - 243.43

Sum of angle = - 243.43+18.43 = - 225

Therefore, the sum of the individual angles z and w is equal to the angle of z. w

|z||w| = √5 * √10=7.07

This product of the individual length of z and w is not equal to the length z. w, this is because the angle between them is 135 assuming the angle is 0 then it will be equal.