An inequality about inner product

In summary, I couldn't find a way to remove the -2Abs(A)Abs(B) from the equation and so the inequality fails to hold.
  • #1
tghg
13
0
If α,β,γ are vectors in the Euclid space V, please show that
|α-β||γ|≤|α-γ||β|+|β-γ||α|,where |α|=√(α,α)
and point out when the equal mark holds.

Can someone help me out?
 
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  • #2
This question reeks of Triangle Inequality, I'm assuming the relation: Abs(A-B) <= Abs(A) + Abs(B)

Abs(A-B)Abs(C) <= Abs(A)Abs(C) + Abs(C)Abs(B)
Abs(A-C)Abs(B) <= Abs(A)Abs(B) + Abs(C)Abs(B)
Abs(B-C)Abs(A) <= Abs(B)Abs(A) + Abs(C)Abs(A)

Abs(C)Abs(B) = Abs(A-C)Abs(B) - Abs(A)Abs(B)
Abs(A)Abs(C) = Abs(B-C)Abs(A) - Abs(B)Abs(A)

Abs(A-B)Abs(C) <= Abs(A-C)Abs(B) + Abs(B-C)Abs(A) - 2Abs(A)Abs(B)

I'm stumped about how to remove the -2Abs(A)Abs(B). :mad:

Equality if A=B=C
 
  • #3
Thanks very much!

Note that 2Abs(A)Abs(B)>=0, so
Abs(A-C)Abs(B) + Abs(B-C)Abs(A) - 2Abs(A)Abs(B) <= Abs(A-C)Abs(B) + Abs(B-C)Abs(A)
thus, Abs(A-B)Abs(C) <= Abs(A-C)Abs(B) + Abs(B-C)Abs(A) - 2Abs(A)Abs(B)
<= Abs(A-C)Abs(B) + Abs(B-C)Abs(A).
It's obvious that if A=B=C the equal mark holds.
But I think there are some other cases that satisfy the the equality.
 
Last edited:
  • #4
How about we go from...

Abs(A-B)Abs(C) <= Abs(A-C)Abs(B) + Abs(B-C)Abs(A) - 2Abs(A)Abs(B)

to basic knowledge about numbers, e.g. 2Abs(A)Abs(B) >= 0, so
Abs(A-C)Abs(B) + Abs(B-C)Abs(A) - 2Abs(A)Abs(B) <= Abs(A-C)Abs(B) + Abs(B-C)Abs(A) which gives you what you want

Back to basics :p
 
  • #5
Oh,my god!I found a severe mistake. We can't claim that
Abs(C)Abs(B) = Abs(A-C)Abs(B) - Abs(A)Abs(B)
Abs(A)Abs(C) = Abs(B-C)Abs(A) - Abs(B)Abs(A)


So, I'm very sorry to say that we didn't verify the inequality.
 
  • #6
The correct relationship is

Abs(C)Abs(B) >= Abs(A-C)Abs(B) - Abs(A)Abs(B)
Abs(A)Abs(C) >= Abs(B-C)Abs(A) - Abs(B)Abs(A)

but ASAICS this leads nowhere. :(
 
Last edited:

1. What is an inner product?

An inner product is a mathematical operation that takes two vectors as inputs and produces a scalar value as output. It is often denoted by <u,v> and is equivalent to the dot product of the two vectors. The result of an inner product measures the similarity or projection of one vector onto another.

2. What is an inequality about inner product?

An inequality about inner product is a statement that compares the results of two inner products using the symbols >, <, , or . These inequalities are used to express relationships between vectors based on their inner products, such as orthogonality or magnitude.

3. How is an inequality about inner product useful?

An inequality about inner product is useful in many areas of mathematics and science, particularly in linear algebra and optimization. It allows us to make general statements about the properties of vectors and their relationships, which can be applied to a wide range of problems and applications.

4. What are some common inequalities about inner product?

Some common inequalities about inner product include the Cauchy-Schwarz inequality, which states that the absolute value of the inner product of two vectors is always less than or equal to the product of their magnitudes, and the Triangle inequality, which states that the magnitude of the sum of two vectors is always less than or equal to the sum of their magnitudes.

5. How do inequalities about inner product relate to real-world problems?

Inequalities about inner product have many real-world applications, such as in physics, engineering, and data analysis. For example, the Cauchy-Schwarz inequality is used in signal processing to measure the similarity between two signals, and the Triangle inequality is used in error analysis to determine the maximum possible error in a measurement based on the errors in its components.

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