Hi, there is a problem on my math book that has been giving me headaches. That's why I'm looking for help. The problem is:
Given z and w, two complex numbers such as |z+w|=|z-w|, prove that
Arg z - arg w = plusminus (pi/2)
In other words, the two vectors are perpendicular
If the two magnitudes are equal, then assuming that |z| >= |w|
|z| = √(a^2+b^2)
|w| = √(c^2+d^2)
z+w = (a+c)+(b+d)i
z-w = (a-c)+(b-d)i
√((a+c)^2+(b+d)^2) = √((a-c)^2+(b-d)^2)
(a+c)^2+(b+d)^2 = (a-c)^2+(b-d)^2
a^2+2ac+c^2 + b^2+2bd+d^2 = a^2-2ac+c^2 + b^2-2bd+d^2
2ac+2bd = -2ac-2bd
ac = -bd
a/b = -d/c
In other words, the slopes of the two vectors are negative reciprocals. The vectors are perpendicular.
sorry - forgot to eliminate my first line of thought; skip that stuff about |z| and |w|.
There's probably a nice complex-number way to do this, but this was my first approach.
Sure, I'd be happy to help you with this math problem!
To prove that Arg z - Arg w = ±π/2, where z and w are complex numbers and |z + w| = |z - w|, we need to make use of some properties of complex numbers and the concept of arguments.
Let's start by understanding what the notation means. The complex number z is usually written as z = a + bi, where a is the real part of z and b is the imaginary part of z. Similarly, w can be written as w = c + di, where c is the real part of w and d is the imaginary part of w.
Now, the absolute value or modulus of z is defined as |z| = √(a^2 + b^2), and the argument of z, denoted as Arg(z), is the angle between the positive real axis and the line segment joining the origin to the point z in the complex plane.
Using these definitions, we can rewrite the given condition as follows:
|z + w| = |z - w|
√((a + c)^2 + (b + d)^2) = √((a - c)^2 + (b - d)^2)
Now, let's simplify this equation step by step:
(a + c)^2 + (b + d)^2 = (a - c)^2 + (b - d)^2
a^2 + 2ac + c^2 + b^2 + 2bd + d^2 = a^2 - 2ac + c^2 + b^2 - 2bd + d^2
4ac + 4bd = 0
ac + bd = 0
At this point, we have ac + bd = 0. It's important to note that ac + bd = 0 implies one of two possibilities: either ac = -bd or ac = bd = 0.
Case 1: ac = -bd
In this case, we can see that either a = b = c = d = 0, or a and c have opposite signs and b and d have the same sign. However, if a = b = c = d = 0, then both z and w are zero, and their arguments are undefined. So, let's consider the second possibility.
For ac = -bd, we can rewrite it as -d/b = c/a. The ratio -d/b represents the tangent of the argument of w, and c/a represents the tangent of the argument of z. Therefore, we have:
Arg(w) = Arg(z) ± π/2
or equivalently,
Arg(z) - Arg(w) = ± π/2
Case 2: ac = bd = 0
In this case, either a = c = 0 or b = d = 0. If a = c = 0, z is purely imaginary, and if b = d = 0, w is purely real. In both cases, their arguments can be anything, and the result Arg(z) - Arg(w) = ± π/2 still holds.
Therefore, in both cases, we have proven that Arg(z) - Arg(w) = ± π/2, under the condition |z + w| = |z - w|.
I hope this explanation helps you understand how to prove the given statement! If you have any further questions, please feel free to ask.