Let z1=2-i,z2=-2+i
Find. Re(z1z2/modulus of z1)
I only know that modulus of z1=
2+i
I have no idea . how to solve this?
the modulus of z1 = |2-i| = √(2^2+1^2) = √5
z1•z2 = (2-i)(-2+i) = -3+4i
so, Re(z1•z2)/√5 = -3/√5
But the answer should be -2/5
In that case, look for a typo somewhere. The numbers you gave produce the given result.
To find Re(z1z2/modulus of z1), we need to first calculate z1z2 and then divide it by the modulus of z1.
Let's start by calculating z1z2:
z1 = 2 - i
z2 = -2 + i
To find z1z2, we can multiply the real and imaginary parts separately:
Real part:
Real(z1z2) = Real(z1) * Real(z2) - Imaginary(z1) * Imaginary(z2)
= (2 * -2) - (-1 * 1)
= -4 + 1
= -3
Imaginary part:
Imaginary(z1z2) = Real(z1) * Imaginary(z2) + Imaginary(z1) * Real(z2)
= (2 * 1) + (-1 * -2)
= 2 + 2
= 4
So, z1z2 = -3 + 4i.
Next, let's calculate the modulus of z1:
modulus of z1 = sqrt(Real(z1)^2 + Imaginary(z1)^2)
= sqrt(2^2 + (-1)^2)
= sqrt(4 + 1)
= sqrt(5)
Now, we can find Re(z1z2/modulus of z1) by dividing the real part of z1z2 by the modulus of z1:
Re(z1z2/modulus of z1) = Real(z1z2) / modulus of z1
= -3 / sqrt(5)
So, Re(z1z2/modulus of z1) = -3 / sqrt(5).
I hope this explanation helps you understand how to solve the problem!