Finding the Length of a Vector using Inner Products and Circles

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In summary, the conversation discusses finding the length of a vector and deriving the equation for a circle based on a given inner product. The equation is found to be 5x^2 + y^2 = 1, with y-intercepts at 1,-1 and x-intercepts at -sqrt(1/5),sqrt(1/5). There is also a discussion about eliminating mixed xy-terms through rotation.
  • #1
TranscendArcu
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Skjermbilde_2012_03_03_kl_1_15_53_PM.png

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 y-intercepts at 1,-1 and x-intercepts at -sqrt(1/5),sqrt(1/5).

Is this correct?
 
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  • #2
I think you're missing some terms from the length:

<(x,y),(x,y)>=5x2+2(xy+yx)+y2
 
  • #3
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
I think if you do a rotation of the plain, you may be able to get rid of the mixed xy-terms.
 

1. What is an inner product?

An inner product is a mathematical operation that takes in two vectors and produces a scalar value. It is used to measure the angle between two vectors and the length of a vector.

2. How is the inner product calculated?

The inner product is calculated by multiplying the corresponding components of two vectors and adding them together. This can also be represented as the dot product of the two vectors.

3. What is the significance of inner products in geometry?

In geometry, inner products are used to calculate the length of a vector, the angle between two vectors, and determine whether two vectors are perpendicular. They also play a crucial role in defining geometric objects such as circles and ellipses.

4. How are inner products related to circles?

Inner products are used to define the concept of orthogonality, which is essential in constructing circles. The radius of a circle can be represented as the length of the perpendicular line from the center of the circle to any point on the circle. This perpendicular line is perpendicular to the tangent line, which is a vector, and the inner product of these two vectors is zero, indicating orthogonality.

5. Can inner products be used in other fields besides geometry?

Yes, inner products have applications in many fields, including physics, engineering, and economics. They are used in linear algebra, functional analysis, and optimization problems. In physics, inner products are used to calculate work, energy, and momentum. In economics, they are used to measure preferences and utility functions.

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