Proof Help: Prove ||x-y|| = ||x|| ||y|| ||x bar - y bar||

  • Thread starter Kuma
  • Start date
  • Tags
    Proof
In summary, the conversation is about a proof involving vectors in Rn and the formula for norm. The person asking for help has tried to solve it mathematically but is now attempting a geometric approach. They are stuck on how to equate the right side of the formula to the left. Another person suggests drawing a triangle and reflecting it to find the same length of the sides. They also mention using an algebraic approach for the proof.
  • #1
Kuma
134
0
Hi, I am having trouble with this proof, I am wondering what to do because the way I have attempted it is incorrect.

I want to prove

||x-y|| = ||x|| ||y|| ||x bar - y bar||

where

x and y are vectors in Rn

and u bar is defined by u/(||u||^2)


so the question asks to prove this analytically, I couldn't figure out how to do so, the only attempt I made was mathematically by expanding the right side using the formula for norm and trying to simplify it to the left. Anyway that results in a mess and I was wondering where to start.
 
Physics news on Phys.org
  • #2
I would draw a picture and do this geometrically. Draw two rays emanating from the origin; these rays are the directions of x and y. Choose a point on each ray to represent each vector. Then ||x-y|| is the length of the line segment that goes between these two points. Can you think of another line segment you can draw between the two rays that obviously has the same length as ||x-y||?
 
  • #3
thanks for the tip. I've started doing it geometrically but I'm still stuck on the right side of the proof and how it equates to the left. The only other ray with the same length should be lly-xll = llx-yll, but I'm still unsure how to end up there.
 
  • #4
You should have a triangle OXY, where O is the origin and X and Y are the endpoints of the vectors x and y. Then ||x-y|| is the length of XY, i.e. the side opposite the angle O.

Now take this picture, and reflect it about the line that bisects angle O. Overlay this on top of your original picture. How do you express the new legs OX' and OY' in terms of the original OX and OY? Then consider that it should be obvious that the legs X'Y' and XY have the same length.
 
  • #5
For intuition, drawing a picture is fine, but for proof, I would use algebraic approach. This formula solves it all:

[tex]
\|A + B\|^2 = \|A\|^2 + \|B\|^2 + 2A\cdot B
[/tex]
 
  • #6
http://www.mathhelpforum.com/math-help/f5/help-proof-189355.html#post685844
 

1. What is the purpose of proving ||x-y|| = ||x|| ||y|| ||x bar - y bar||?

The purpose of proving this equation is to show the relationship between the absolute values of two vectors, x and y, and their conjugates. It helps to understand the properties of vector magnitude and how it relates to complex numbers.

2. What are the steps involved in proving this equation?

The steps involved in proving this equation may vary depending on the approach taken. However, a common approach would involve using the definition of vector magnitude, properties of complex numbers, and algebraic manipulation to show that both sides of the equation are equal.

3. Can this equation be proven using any type of vector or does it only apply to complex numbers?

This equation can be proven for any type of vector, not just complex numbers. However, the properties of complex numbers are often used in the proof, making it easier to understand and apply.

4. What are the implications of this equation in the field of mathematics or science?

This equation has implications in various fields of mathematics and science, such as physics, engineering, and computer science. It helps to understand the behavior of vectors and complex numbers, which are used in many mathematical and scientific models and calculations.

5. Are there any real-world applications of this equation?

Yes, there are many real-world applications of this equation. For example, in physics, it can be used to analyze the electric and magnetic fields in electromagnetic waves. In engineering, it can be used to design circuits and analyze the behavior of electrical components. In computer science, it can be used in image and signal processing algorithms. Overall, this equation has numerous practical applications in various fields.

Similar threads

Replies
3
Views
1K
  • Linear and Abstract Algebra
Replies
1
Views
816
  • Calculus and Beyond Homework Help
Replies
2
Views
515
Replies
5
Views
1K
  • Linear and Abstract Algebra
Replies
2
Views
1K
  • Precalculus Mathematics Homework Help
Replies
10
Views
1K
  • Linear and Abstract Algebra
Replies
20
Views
3K
  • Calculus and Beyond Homework Help
Replies
20
Views
2K
Replies
4
Views
253
  • Calculus and Beyond Homework Help
Replies
32
Views
2K
Back
Top