# Using the condition of vector equilibrium

## Homework Statement

Using the condition of vector equilibrium, prove that
$$\sum$$$$^{2n-1}_{k=0}$$ 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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dx
Homework Helper
Gold Member
The sigma is a symbol for summation. For example

$$\sum_{i = 1}^{i = 5} f(i)$$

just means $$f(1) + f(2) + f(3) + f(4) + f(5)$$. The subscript is the value at which i begins, and the superscript is the value at which i ends.

tiny-tim
Homework Helper
I know that if n equal vectors are at an angle (2*pi)/n
Their resultant is zero.
HI ritwik06!

Hint: Draw those n equal vectors.

What are the coordinates of their endpoints?

Using the condition of vector equilibrium, prove that:

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

Hey, help me please. Please tell me what do I have to prove.

Last edited:
dx
Homework Helper
Gold Member
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.

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

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!

dx
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Gold Member
No, its not. n is the number of vectors. Its not changing.

dx
Homework Helper
Gold Member
Thir sum will be zero!
Ok, so write it out explicitly and see what its gonna look like.