- #1
Yichen
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- quadratic equation ||v||^2 - c(2v·w)+c^2||w||^2=0, where c belongs to any real number, v and w are both vectors
What is the discriminant of the following quadratic equation
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SUMMARY
The discussion centers on the quadratic equation ||v||^2 - c(2v·w) + c^2||w||^2 = 0, where c is a real number and v and w are vectors. The equation can be transformed into the form $$\left ({\bf \vec v} - c {\bf \vec w } \right ) \cdot \left ({\bf \vec v} - c {\bf \vec w } \right ) = 0$$, indicating that a solution exists where {\bf \vec v} = c {\bf \vec w}. Additionally, there is a suggestion that the discussion may relate to proving the Cauchy-Schwarz inequality.
PREREQUISITES- Understanding of quadratic equations in vector spaces
- Familiarity with vector operations, including dot products
- Knowledge of the Cauchy-Schwarz inequality
- Basic algebraic manipulation skills
- Study the properties of quadratic equations in vector spaces
- Learn about vector dot products and their geometric interpretations
- Research the proof and applications of the Cauchy-Schwarz inequality
- Explore methods for solving quadratic equations involving vectors
Students studying linear algebra, mathematicians interested in vector calculus, and anyone looking to understand the applications of quadratic equations in higher mathematics.
- #2
BvU
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Hello Yichen, 
Please post in the homework forum. There is a most useful template there for questions like this.
If this is an equation, what are the knowns and what is the unknown ?
I read your equation as equivalent to $$\left ({\bf \vec v} - c {\bf \vec w } \right ) \cdot \left ({\bf \vec v} - c {\bf \vec w } \right ) = 0 $$ which certainly has a solution ##\ {\bf \vec v} = c {\bf \vec w } ##
Please post in the homework forum. There is a most useful template there for questions like this.
If this is an equation, what are the knowns and what is the unknown ?
I read your equation as equivalent to $$\left ({\bf \vec v} - c {\bf \vec w } \right ) \cdot \left ({\bf \vec v} - c {\bf \vec w } \right ) = 0 $$ which certainly has a solution ##\ {\bf \vec v} = c {\bf \vec w } ##
- #3
pasmith
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Yichen said:
Are you attempting to prove the Cauchy-Schwarz inequality?
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