Question

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

Answer 1

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.