What is geometric interpretation of this equality?

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SUMMARY

The discussion centers on proving the equality |a - b|² + |a + b|² = 2(|a|² + |b|²) for any complex numbers a and b. Participants explore the geometric interpretation of this equality, identifying that |a - b| and |a + b| represent the lengths of the diagonals of a parallelogram formed by vectors a and b in the complex plane. The conversation emphasizes the need to express squared terms as lengths and suggests that the equality relates to the properties of parallelograms in geometry.

PREREQUISITES
  • Understanding of complex numbers and their properties
  • Familiarity with the concept of the Argand diagram
  • Knowledge of vector representation in geometry
  • Basic understanding of parallelogram properties and the Pythagorean theorem
NEXT STEPS
  • Study the geometric properties of parallelograms and their diagonals
  • Learn about the Argand diagram and its applications in complex number visualization
  • Research the relationship between complex numbers and vector algebra
  • Explore plane geometry theorems related to the squares of the diagonals of a parallelogram
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Mathematics students, educators, and anyone interested in the geometric interpretation of complex number relationships and their applications in algebra and geometry.

Shackleford
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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. -_-
 
Last edited:
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I believe you can just interpret this as twice the distance from points a and b in the complex plane.
 
Shackleford said:

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?
 
Mark44 said:
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.
 
Shackleford said:
Eh. The short diagonal of parallelogram.
Yep.
 
Mark44 said:
Yep.

So we're taking the norm of those two vectors, squaring them, and adding them.
 
Is it some sort of rotation and enlarging of the parallelogram?
 
Shackleford said:
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?
 
haruspex said:
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.
 
  • #10
Shackleford said:
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).
 
  • #11
haruspex said:
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. o0):headbang:
 
  • #12
Shackleford said:
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. o0):headbang:
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.
 
  • #13
haruspex said:
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.
 
  • #14
Shackleford said:
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?
 
  • #15
haruspex said:
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?
 
  • #16
Shackleford said:
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.)
 
  • #17
haruspex said:
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.
 
  • #18
Shackleford said:
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|?
 
  • #19
haruspex said:
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.
 
  • #20
Shackleford said:
Yes, I see the parallelogram and the sides.
So express the equation in terms of the lengths PQ etc.
 
  • #21
haruspex said:
So express the equation in terms of the lengths PQ etc.

(PR)2 +(PS)2 = 2[(PQ)2 + (RS)2]
 
  • #22
Shackleford said:
(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?
 
  • #23
haruspex said:
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.
 
  • #24
Shackleford said:
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.
 
  • #25
haruspex said:
No, that's wrong. One of the vertices should be the origin.

(PT)2 +(PR)2 = 2[(PS)2 + (PQ)2]
 
  • #26
Shackleford said:
(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.
 
  • #27
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.
 

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