Using the condition of vector equilibrium

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Homework Help Overview

The discussion revolves around proving a summation involving cosine terms, specifically the expression \(\sum^{2n-1}_{k=0} \cos \left(\frac{k \pi}{n}\right)\), where \(n\) is an integer. The context is rooted in vector equilibrium and the properties of vectors at equal angles.

Discussion Character

  • Exploratory, Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants are attempting to clarify the meaning of the summation notation and its components. There are discussions about the implications of vector components and their resultant sum. Some participants suggest visualizing the vectors to aid understanding.

Discussion Status

The discussion is ongoing, with participants providing hints and asking for clarifications. There is a focus on understanding the components of the vectors and their relationship to the summation. Multiple interpretations of the problem are being explored, with no explicit consensus reached yet.

Contextual Notes

Participants are navigating potential misunderstandings regarding the notation and the nature of the summation limits. There is an emphasis on ensuring clarity about the variables involved and their roles in the problem.

ritwik06
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Homework Statement


Using the condition of vector equilibrium, prove that
[tex]\sum[/tex][tex]^{2n-1}_{k=0}[/tex] cos ((k*pi)/n)

Where n is an integer.

Homework Equations



I know that if n equal vectors are at an angle (2*pi)/n
Their resultant is zero.

The Attempt at a Solution



Please first explain to me what does this sigma signify and what is meant by its subscript and superscript. Then please give me an idea of what to do with this problem. Thanks.

Ritwik
 
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The sigma is a symbol for summation. For example

[tex]\sum_{i = 1}^{i = 5} f(i)[/tex]

just means [tex]f(1) + f(2) + f(3) + f(4) + f(5)[/tex]. The subscript is the value at which i begins, and the superscript is the value at which i ends.
 
ritwik06 said:
I know that if n equal vectors are at an angle (2*pi)/n
Their resultant is zero.

HI ritwik06! :smile:

Hint: Draw those n equal vectors.

What are the coordinates of their endpoints? :smile:
 
Using the condition of vector equilibrium, prove that:

[tex]\sum_{k = 0}^{2n-1} cos \frac{k*pi}{n}=0[/tex]
where n is an integer

Hey, help me please. Please tell me what do I have to prove.
Hey dx. I agree with your format but i don't think it is applicable with my form of question. Please help!
 
Last edited:
Think about the components of the vectors you mentioned in your first post. What are the components, and what should their sum be?

ritwik06 said:
Hey dx. I agree with your format but i don't think it is applicable with my form of question. Please help!

What do you mean its not applicable? All I did was show you what sigma meant, because you asked.
 
Precaution:
cos is upon the whole fraction {(k*pi)/n}
Please just let me know once what I need to prove. As u said dx that the superscript signifis the upperlimit of the variable. but in this case the limit itself is a variable. Please help
 
dx said:
Think about the components of the vectors you mentioned in your first post. What are the components, and what should their sum be?



What do you mean its not applicable? All I did was show you what sigma meant, because you asked.

Thir sum will be zero!

Oh, yes but please re chck the format of my question. Please!
 
No, its not. n is the number of vectors. Its not changing.
 
ritwik06 said:
Thir sum will be zero!

Ok, so write it out explicitly and see what its going to look like.
 

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