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Inner products and Circles 
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#1
Mar412, 10:06 AM

P: 288

Mostly I'd like to look at the third part of the problem. I'm not sure if this is the correct way to derive the equation: So, finding the length of a given vector given this inner product: [itex]<(x,y),(x,y)> = 5x^2 + y^2[/itex]. Taking the length, we have [itex](x,y) = \sqrt{5x^2 + y^2}[/itex], which we define as equaling 1. Squaring both sides we find, [itex]5x^2 + y^2 = 1[/itex]. I think this is the equation of the circle, but I'm not sure. If it is, then my picture has yintercepts at 1,1 and xintercepts at sqrt(1/5),sqrt(1/5). Is this correct? 


#2
Mar412, 10:25 AM

Sci Advisor
P: 1,169

I think you're missing some terms from the length:
<(x,y),(x,y)>=5x^{2}+2(xy+yx)+y^{2} 


#3
Mar412, 10:46 AM

P: 288

Whoops. You're right. My real equation is [itex]5x^2 2(xy+xy) +y^2 =1[/itex]. This changes shape of the circle (it's more elongated in quadrants I and III now), but the intercepts remain the same I think. No?



#4
Mar412, 09:58 PM

Sci Advisor
P: 1,169

Inner products and Circles
I think if you do a rotation of the plain, you may be able to get rid of the mixed xyterms.



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