Proving Inequality with Cosine Rule and Schwartz Proof

In summary: You could show that the norm of the vector is less than 1 by showing that the square of the vector is negative.In summary, our teacher used the schwartz inequality to prove that the expression \frac{a.b}{\|{a}\|\|{b}\|}} lies within the domain [-1,1].
  • #1
SeReNiTy
170
0
Hi, i was required to show that

-1 < [tex] \frac{a.b}{\|{a}\|\|{b}\|}} [/tex] > -1

I did this by using the cosine rule which is [tex] c^2 = a^2 + b^2 - 2a.b\cos{\vartheta} [/tex]

How ever our teacher did it by a scharts proof which i don't quite understand, :mad: , Now my question is why can't i prove it using the cosine rule and could somebody explain the schwartz proof a bit better?
 
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  • #2
What is going on with your inequality. It looks like you have -1 < something > -1. Why not just say that -1 < something. Also, how are you defining a.b? Is it the regular dot product a.b = |a||b|cos(theta)? If so, then this just says cos(theta) > -1 which is false, it may be -1 (if theta = pi).
 
  • #3
SeReNiTy said:
Hi, i was required to show that

-1 < [tex] \frac{a.b}{\|{a}\|\|{b}\|}} [/tex] > -1

I did this by using the cosine rule which is [tex] c^2 = a^2 + b^2 - 2a.b\cos{\vartheta} [/tex]

How ever our teacher did it by a scharts proof which i don't quite understand, :mad: , Now my question is why can't i prove it using the cosine rule and could somebody explain the schwartz proof a bit better?
Has your teacher defined the angle between two vectors to be the arccosine of your expression, or is he using the inequality 1 > [tex] \frac{a.b}{\|{a}\|\|{b}\|}} [/tex] > -1 to motivate such a definition for general vector spaces (not just Euclidean space) ? In the latter case, it is understandable that he does not equate it with what he is trying to motivate yet.
 
  • #4
yes the inequality is suppose to be -1< something < 1

Also I'm suppose to prove that the expression [tex] \frac{a.b}{\|{a}\|\|{b}\|}} [/tex]
lies within the domain [-1,1]

I did this using the cosine rule because cos(theta) is within that domain, how do i do it with the schwartz proof?
 
  • #5
Schwarz's Inequality states that (a.b)² < (a.a)(b.b) does it not? And you've defined the norm by |a|² = a.a, right? You're really not asking the question right because you haven't shown your work, you haven't shown what you know, etc. Anyways, assuming that you have the above inequality (Schwarz's) and definition (of norm) to work with, isn't it obvious?
 

1. How is the cosine rule used to prove inequalities?

The cosine rule, also known as the law of cosines, is a mathematical theorem that relates the sides and angles of a triangle. It states that the square of one side of a triangle is equal to the sum of the squares of the other two sides minus two times the product of those two sides and the cosine of the included angle. This rule can be used to prove inequalities by comparing the values of the sides and angles of two different triangles.

2. What is the Schwartz proof and how is it used in proving inequalities?

The Schwartz proof is a geometric proof that uses the concept of similarity to prove inequalities. It involves creating a similar triangle to the original triangle and using the properties of similar triangles to show that the sides and angles of the original triangle are in fact unequal. This proof is often used in conjunction with the cosine rule to provide a more comprehensive proof of an inequality.

3. Can the cosine rule and Schwartz proof be used for all types of triangles?

Yes, the cosine rule and Schwartz proof can be used for all types of triangles, including acute, obtuse, and right triangles. However, the specific approach and calculations may vary depending on the type of triangle being analyzed.

4. What are some practical applications of proving inequalities with the cosine rule and Schwartz proof?

The cosine rule and Schwartz proof are often used in fields such as engineering, physics, and geometry to solve real-world problems involving triangles. For example, they can be used to determine the stability of structures, calculate distances and heights, and analyze the forces acting on objects.

5. Are there any limitations to using the cosine rule and Schwartz proof in proving inequalities?

One limitation of using the cosine rule and Schwartz proof is that they require a good understanding of geometry and trigonometry concepts. Additionally, these methods may not be applicable to more complex shapes or situations where the triangle being analyzed is not well defined. In these cases, other mathematical or analytical methods may need to be used to prove inequalities.

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