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

sam0617
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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.
 
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What is <u,v> and ||u|| in \mathbb{R}^2??

Can you rewrite the Cauchy-Schwarz inequality in \mathbb{R}^2??
 
micromass said:
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.
 
sam0617 said:
<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??
 
micromass said:
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
 
Can you write out

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

??

Just write out what <u,v> and ||u|| mean...
 
micromass said:
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)
 
Well, thanks anyway. I got help from the wonderful people at yahoo answers. Thank you for trying to work with me.
 
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