Describing A Mathematical Result

In summary, the given equation states that the square of the magnitude of a vector b is equal to the square of the dot product between b and an arbitrary unit vector u, plus the square of the cross product between the two vectors. This can be understood as the magnitude of a vector \vec{b_{\vec{u}}} that goes in the direction of u, represented by the dot product, and a vector \vec{b_{||}} that is orthogonal to u, represented by the cross product. The statements given in the post are valid, and the first term on the right-hand side should be squared.
  • #1
Bashyboy
1,421
5

Homework Statement


Let u be an arbitrary fixed unit vector and show that an vector b satisfies [itex]b^2 = (\vec{u} \cdot \vec{b}) + (\vec{u} \times \vec{b})^2[/itex] Explain this result in words, with the help of a picture.

Homework Equations


The Attempt at a Solution


I understand that the equations says that the square of the magnitude of some vector b is equal to the square of the dot product of b and some arbitrary unit vector u, plus the square of the cross product between the two vectors alluded to already.

I want to examine the dot product first. [itex]\vec{u} \cdot \vec{b} = |u||b|\cos \theta[/itex]. Is it correct to state that the cross product represents the amount of vector b that goes (points) in the direction of vector u. So, the right side of the equation can be thought of the magnitude of some vector[itex]\vec{b_{\vec{u}}}[/itex], such that [itex]\vec{b_{\vec{u}}} = c \vec{u}[/itex], and [itex]\vec{b} = \vec{b_{\vec{u}}} + \vec{b_{||}}[/itex], where [itex]\vec{b_{||}}[/itex] is orthogonal to the vector u.

Are these correct statements?
 
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  • #2
Can you write the question as it is supposed to be written.
 
  • #3
The description given in section 1 is the exact problem.
 
  • #4
Bashyboy said:
I want to examine the dot product first. [itex]\vec{u} \cdot \vec{b} = |u||b|\cos \theta[/itex]. Is it correct to state that the cross product represents the amount of vector b that goes (points) in the direction of vector u. So, the right side of the equation can be thought of the magnitude of some vector[itex]\vec{b_{\vec{u}}}[/itex], such that [itex]\vec{b_{\vec{u}}} = c \vec{u}[/itex], and [itex]\vec{b} = \vec{b_{\vec{u}}} + \vec{b_{||}}[/itex], where [itex]\vec{b_{||}}[/itex] is orthogonal to the vector u.

Are these correct statements?

I simply want to know if these statements are valid.
 
  • #5
Are you sure that the first term on the rhs is [itex](\vec{u} \cdot \vec{b})[/itex] rather than [itex](\vec{u} \cdot \vec{b})^2[/itex]?
 
  • #6
Chester, you are correct. It should be squared.
 
  • #7
So, am I to assume the statements I quoted in post #4 are correct, as no one has opposed them?
 
  • #8
[itex](\vec{u} \cdot \vec{b})^2=b^2\cos^2{\theta}[/itex]
[itex](\vec{u} \times \vec{b})\cdot(\vec{u} \times \vec{b})=b^2\sin^2{\theta}[/itex]
 

FAQ: Describing A Mathematical Result

What is the purpose of describing a mathematical result?

The purpose of describing a mathematical result is to communicate the findings and conclusions of a mathematical study or experiment to others in a clear and concise manner. This allows for the replication of the study by others and contributes to the overall knowledge and understanding of the mathematical concept being studied.

How should a mathematical result be described?

A mathematical result should be described using precise and specific language, including numerical values, equations, and relevant statistical analysis. It is important to use appropriate mathematical notation and terminology to accurately convey the findings of the study.

What should be included in the description of a mathematical result?

The description of a mathematical result should include a clear statement of the research question or hypothesis, the methods and procedures used, the data collected, and the results and conclusions drawn from the analysis. It may also include any limitations or areas for further research.

How should the results of a mathematical study be presented?

The results of a mathematical study should be presented in a logical and organized manner, using tables, graphs, and figures to visually represent the data. The presentation should also include a written explanation of the results and their significance in relation to the research question or hypothesis.

Why is it important to accurately describe a mathematical result?

Accurately describing a mathematical result is important because it allows for the replication and verification of the study by others, which contributes to the overall validity and reliability of the findings. It also allows for the integration of the results into the existing body of knowledge within the field of mathematics.

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