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

[sam0617]

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.

 

[micromass]

What is <u,v> and ||u|| in \mathbb{R}^2??

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

 

[sam0617]

What is <u,v> and ||u|| in \mathbb{R}^2??

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

<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.

 

[micromass]

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

Good

and ||u|| in R² is √u²

No, that would be \sqrt{u_1^2+u_2^2}.

Anybody, do you recognize something like u_1v_1+u_2v_2 in your OP??

 

[sam0617]

Good



No, that would be \sqrt{u_1^2+u_2^2}.

Anybody, do you recognize something like u_1v_1+u_2v_2 in your OP??

I see that a² + b² would be
a₁²a₁²+b₁²b₁²

 

[micromass]

Can you write out

&lt;u,v&gt;^2\leq \|u\|^2\|v\|^2

??

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

 

[sam0617]

Can you write out

&lt;u,v&gt;^2\leq \|u\|^2\|v\|^2

??

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)

 

[sam0617]

Well, thanks anyway. I got help from the wonderful people at yahoo answers. Thank you for trying to work with me.