What is geometric interpretation of this equality?

[Shackleford]

Homework Statement

Prove that for any a, b ∈ ℂ, |a - b|² + |a + b|² = 2(|a|² + |b|²).

Homework Equations

|a|² = aa*

(a - b)* = (a* - b*)
(a + b)* = (a* + b*)

* = complex conjugate

The Attempt at a Solution

I've already shown that the relation is true. I'm not quite sure what the geometric interpretation is. My guess is a side of a triangle. -_-

 

[MostlyHarmless]

I believe you can just interpret this as twice the distance from points a and b in the complex plane.

 

[Mark44]

Homework Statement

Prove that for any a, b ∈ ℂ, |a - b|² + |a + b|² = 2(|a|² + |b|²).

Homework Equations

|a|² = aa*

(a - b)* = (a* - b*)
(a + b)* = (a* + b*)

* = complex conjugate

The Attempt at a Solution

I've already shown that the relation is true. I'm not quite sure what the geometric interpretation is. My guess is a side of a triangle. -_-

Well, you're warm. If a and be represent vectors that are two adjacent sides of a parallelogram, a + b is a vector that represents the long diagonal of the parallelogram. Can you figure out what a - b represents?

 

[Shackleford]

Well, you're warm. If a and be represent vectors that are two adjacent sides of a parallelogram, a + b is a vector that represents the long diagonal of the parallelogram. Can you figure out what a - b represents?

Eh. The short diagonal of parallelogram.

 

[Mark44]

Eh. The short diagonal of parallelogram.

Yep.

 

[Shackleford]

Yep.

So we're taking the norm of those two vectors, squaring them, and adding them.

 

[Shackleford]

Is it some sort of rotation and enlarging of the parallelogram?

 

[haruspex]

Is it some sort of rotation and enlarging of the parallelogram?

No.
Are you seeking a geometrical description for your own interest, or is it part of the question? If the second, is it supposed to match some named theorem?

 

[Shackleford]

No.
Are you seeking a geometrical description for your own interest, or is it part of the question? If the second, is it supposed to match some named theorem?

It's the last half of the question. I'm not quite sure. I don't see anything in the notes that the professor provides for the class.

 

[haruspex]

It's the last half of the question. I'm not quite sure. I don't see anything in the notes that the professor provides for the class.

Ok. So the first step (and perhaps the only step) is to express each of the squared entities as a length. You already have |a-b|, |a+b|. The other two are easier, except that there is a small ambiguity (which you can exploit to make the geometric statement symmetric).

 

[Shackleford]

Ok. So the first step (and perhaps the only step) is to express each of the squared entities as a length. You already have |a-b|, |a+b|. The other two are easier, except that there is a small ambiguity (which you can exploit to make the geometric statement symmetric).

I've already shown algebraically that the relation is true, using the given equations. That was fairly straightforward. I'm just not (and perhaps I should) seeing the geometric interpretation.

 

[haruspex]

I've already shown algebraically that the relation is true, using the given equations. That was fairly straightforward. I'm just not (and perhaps I should) seeing the geometric interpretation.

Yes, I understand that, and my post was intended to assist with that. Express each of the squared entities as a length within the diagram. You already have |a-b|, |a+b| as lengths of diagonals of the parallelogram. |a| and |b| are even easier. The only vaguely interesting bit is interpreting the 2 in not quite the obvious way.

 

[Shackleford]

Yes, I understand that, and my post was intended to assist with that. Express each of the squared entities as a length within the diagram. You already have |a-b|, |a+b| as lengths of diagonals of the parallelogram. |a| and |b| are even easier. The only vaguely interesting bit is interpreting the 2 in not quite the obvious way.

If I'm not mistaken, the aa* and bb* are vectors mapped to the real axis with lengths of a and b squared, respectively.

 

[haruspex]

If I'm not mistaken, the aa* and bb* are vectors mapped to the real axis with lengths of a and b squared, respectively.

Mapped to the real axis? As in rotated?
OK, but what are they in terms of the geometry. Don't mention vectors in answering that. They're just squares of lengths of... what?

 

[Shackleford]

Mapped to the real axis? As in rotated?
OK, but what are they in terms of the geometry. Don't mention vectors in answering that. They're just squares of lengths of... what?

You add the angles of the vector and its complex conjugate and the norm is the quantity squared. A square?

 

[haruspex]

You add the angles of the vector and its complex conjugate and the norm is the quantity squared. A square?

Did you draw a diagram of the vectors and of the parallelogram that Mark44 referred to?
(Actually, because this is stated in terms of complex numbers, not vectors, we should be discussing an Argand diagram, but it will come to the same thing.)

 

[Shackleford]

Did you draw a diagram of the vectors and of the parallelogram that Mark44 referred to?
(Actually, because this is stated in terms of complex numbers, not vectors, we should be discussing an Argand diagram, but it will come to the same thing.)

I just redrew it. I'm seeing a few geometric shapes pop out.

 

[haruspex]

I just redrew it. I'm seeing a few geometric shapes pop out.

OK, but you see the parallelogram Mark44 mentioned? Let the vertices in cyclic order be P, Q, R, S. Do you see what distances in that have lengths |a|, |b|, |a-b|, |a+b|?

 

[Shackleford]

OK, but you see the parallelogram Mark44 mentioned? Let the vertices in cyclic order be P, Q, R, S. Do you see what distances in that have lengths |a|, |b|, |a-b|, |a+b|?

Yes, I see the parallelogram and the sides.

 

[haruspex]

Yes, I see the parallelogram and the sides.

So express the equation in terms of the lengths PQ etc.

 

[Shackleford]

So express the equation in terms of the lengths PQ etc.

(PR)² +(PS)² = 2[(PQ)² + (RS)²]

 

[haruspex]

(PR)² +(PS)² = 2[(PQ)² + (RS)²]

There are a couple of mistakes there, probably just typos.
On the right hand side, instead of having that factor 2 there, can you see how to include more distances and hence make it more symmetric?

 

[Shackleford]

There are a couple of mistakes there, probably just typos.
On the right hand side, instead of having that factor 2 there, can you see how to include more distances and hence make it more symmetric?

Typos? In the positive circular direction, my vertices are P, Q, R, S corresponding to b, a+b, a, and a-b. I guess that I could've used QR instead of RS.

 

[haruspex]

my vertices are P, Q, R, S corresponding to b, a+b, a, and a-b

No, that's wrong. One of the vertices should be the origin.

 

[Shackleford]

No, that's wrong. One of the vertices should be the origin.

(PT)² +(PR)² = 2[(PS)² + (PQ)²]

 

[haruspex]

(PT)² +(PR)² = 2[(PS)² + (PQ)²]

The right hand side is ok now (though as I wrote, it could be expressed more symmetrically. What other distances are equal to PQ and PS?)
But the left hand side is still wrong. Where's T?
Tell me how you're mapping the origin and the complex numbers a and b to P, Q, R S.

 

[Stephen Tashi]

A key issue is whether the "well known" geometric result is really well known. Look on the web for plane geometry theorems about the the squares of the diagonals of a parallelogram.