Using the Cauchy-Schwarz inequality to prove all real values for a, b, and theta

In summary, the conversation discusses the use of the Cauchy-Schwarz inequality to prove that for all real values of a, b, and theta, (a cosθ + b sinθ)2 ≤ a2 + b2. The conversation also clarifies the notation <u,v> and ||u|| in R2 and how to rewrite the inequality in this context. Eventually, the conversation concludes with the correct solution being provided by another source.
  • #1
sam0617
18
1

Homework Statement


Use the Cauchy-Schwarz inequality to prove that for all real values of a, b, and theta (which ill denote as θ),
(a cosθ + b sinθ)2 ≤ a2 + b2



Homework Equations


so the Cauchy-Schwarz inequality is | < u,v>| ≤ ||u|| ||v||



The Attempt at a Solution



I'm having a difficult time figuring out what is my u and my v. Sorry for the stupid question.

Thank you for any guidance.
 
Last edited:
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  • #2
What is <u,v> and ||u|| in [itex]\mathbb{R}^2[/itex]??

Can you rewrite the Cauchy-Schwarz inequality in [itex]\mathbb{R}^2[/itex]??
 
  • #3
micromass said:
What is <u,v> and ||u|| in [itex]\mathbb{R}^2[/itex]??

Can you rewrite the Cauchy-Schwarz inequality in [itex]\mathbb{R}^2[/itex]??

<u,v> in R2 would be u1v1 + u2v2 I believe..

and ||u|| in R2 is √u2

I'm terribly sorry but I'm not sure what you're trying to get at. I know and I also don't want to be spoon fed the answer.
 
  • #4
sam0617 said:
<u,v> in R2 would be u1v1 + u2v2 I believe..

Good

and ||u|| in R2 is √u2

No, that would be [itex]\sqrt{u_1^2+u_2^2}[/itex].

Anybody, do you recognize something like [itex]u_1v_1+u_2v_2[/itex] in your OP??
 
  • #5
micromass said:
Good



No, that would be [itex]\sqrt{u_1^2+u_2^2}[/itex].

Anybody, do you recognize something like [itex]u_1v_1+u_2v_2[/itex] in your OP??

I see that a2 + b2 would be
a12a12+b12b12
 
  • #6
Can you write out

[tex]<u,v>^2\leq \|u\|^2\|v\|^2[/tex]

??

Just write out what <u,v> and ||u|| mean...
 
  • #7
micromass said:
Can you write out

[tex]<u,v>^2\leq \|u\|^2\|v\|^2[/tex]

??

Just write out what <u,v> and ||u|| mean...

Sorry, I think you're losing patience with me but I did what you asked and here it is.
Please correct me if I'm incorrect.

<u,v>2≤∥u∥2∥v∥2
= (u1v1+u2
v2)(u1v1+u2
v2) ≤ √(u12+ u22) √(v12 + v22)
 
  • #8
Well, thanks anyway. I got help from the wonderful people at yahoo answers. Thank you for trying to work with me.
 

FAQ: Using the Cauchy-Schwarz inequality to prove all real values for a, b, and theta

1. What is the Cauchy-Schwarz inequality and how is it used?

The Cauchy-Schwarz inequality is a mathematical inequality that states that for any two vectors in an inner product space, the square of the dot product of the two vectors is always less than or equal to the product of the square of the norm of each vector. This inequality is commonly used in proving theorems in mathematics and is especially useful in the study of real numbers.

2. How does the Cauchy-Schwarz inequality prove all real values for a, b, and theta?

The Cauchy-Schwarz inequality can be used to prove that the dot product of two vectors is always less than or equal to the product of their norms. By setting the vectors to be the two sides of a triangle with sides a and b and the angle between them as theta, we can use the inequality to prove that the length of the third side (c) must be a real number, thus proving the existence of real values for a, b, and theta.

3. Can the Cauchy-Schwarz inequality be used for any type of triangle?

Yes, the Cauchy-Schwarz inequality can be used for any type of triangle, as long as the vectors used in the inequality are the two sides of the triangle and the angle between them. This is because the inequality is a general statement about the relationship between the dot product and the norms of vectors, and is not limited to any specific type of triangle.

4. Are there any other applications of the Cauchy-Schwarz inequality?

Yes, the Cauchy-Schwarz inequality has many applications in mathematics, physics, and other sciences. It is commonly used in optimization problems, statistics, and probability theory. It also has applications in geometry, functional analysis, and linear algebra.

5. Is the Cauchy-Schwarz inequality always true?

Yes, the Cauchy-Schwarz inequality is always true and is considered to be one of the fundamental inequalities in mathematics. It can be proven using various methods, including the Cauchy-Schwarz proof, the geometric proof, and the algebraic proof. Its validity is not limited to any specific set of numbers or vectors, making it a powerful tool in mathematical proofs and applications.

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