Question: If z1 , z2 ,z3 are complex numbers and |z1|=|z2|=|z3|=1 and (z1)^3+(z2)^3 + (z3)^3 +z1*z2*z3=0 find the maximum and minimum values of |z1+z2+z3|.
My thoughts on the question:
Let z1=a , z2=b and z3=
We know that |a+b+c| <= |a| + |b+c|
|a|=1
So |a+b+c|<= 1 + |b+c|
|b+c|<=|b|+ |c|
|b+c|<= 2
So |a+b+c|<= 3
|a+b+c|max=3
Above is how I first solved this.But as I had another look at the given equation I realized that taking the magnitudes z1,z2,z3 all as 1 ,violates the given equation.And also realized that the equation is true only when magnitudes two of the three complex numbers are 1.
So what should be the maximum and the minimum?
I'm looking for a direct method except from plugging in values and getting the result.
To find the maximum and minimum values of |z1+z2+z3|, we need to make use of the given equation: (z1)^3 + (z2)^3 + (z3)^3 + z1*z2*z3 = 0.
Note that the given equation is symmetric with respect to z1, z2, and z3. So, without loss of generality, let's assume that |z1| = |z2| = 1 and |z3| < 1.
Now, let's express z1, z2, and z3 in exponential form: z1 = e^(iθ1), z2 = e^(iθ2), and z3 = e^(iθ3), where θ1, θ2, and θ3 are the arguments of z1, z2, and z3, respectively.
Using the fact that |z1| = |z2| = 1, we can rewrite the given equation as:
e^(3iθ1) + e^(3iθ2) + e^(3iθ3) + e^(i(θ1 + θ2 + θ3)) = 0.
Now, let's write this equation in terms of real and imaginary parts. Since e^(ix) = cos(x) + i*sin(x), we have:
cos(3θ1) + cos(3θ2) + cos(3θ3) + cos(θ1 + θ2 + θ3) + i*(sin(3θ1) + sin(3θ2) + sin(3θ3) + sin(θ1 + θ2 + θ3)) = 0.
Since the equation is satisfied for all values of θ1, θ2, and θ3, the real and imaginary parts of the equation must be zero individually.
Setting the real part equal to zero:
cos(3θ1) + cos(3θ2) + cos(3θ3) + cos(θ1 + θ2 + θ3) = 0.
Setting the imaginary part equal to zero:
sin(3θ1) + sin(3θ2) + sin(3θ3) + sin(θ1 + θ2 + θ3) = 0.
Now, we can see that if two of the θ's are equal, say θ1 = θ2, then the remaining equation becomes:
cos(3θ1) + cos(3θ3) + cos(3θ3) + cos(2θ1 + θ3) = 0.
Since cos(x) + cos(y) + cos(z) + cos(x + y + z) = 4*cos((x + y + z)/2)*cos((x - y)/2)*cos((x - z)/2)*cos((y - z)/2), we can simplify the above equation to:
4*cos((2θ1 + θ3)/2)*cos((2θ1 - θ3)/2) = 0.
This gives us two cases:
1. cos((2θ1 + θ3)/2) = 0:
In this case, the value of |z1 + z2 + z3| is at its maximum, which is 3.
2. cos((2θ1 - θ3)/2) = 0:
In this case, the value of |z1 + z2 + z3| is at its minimum, which is 1.
Therefore, the maximum value of |z1 + z2 + z3| is 3, and the minimum value is 1.