Finding length of vector with unknown variable

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1. Dec 21, 2016

cathal84

• Member warned that the homework template is required
Finding length of vector with unknown variable.
Purely for study purposes.
Find the smallest possible length of the vector →v .
Let vector V = (-2/3,b,16/7)

Equation for finding length of vector : Sqrt(a^2+b^2+c^2)

Question would be quite straight forward had there been no unknown variable but since there is i am quite stuck.
when i work out the equation as per normal i get Sqrt(50/21+b^2)
Is it possible to work out the unknown variable in this question? I don't think it is but please correct me if I'm wrong

2. Dec 21, 2016

Replusz

That is when the sqrt(a^2+b^2+c^2) is minimal.

3. Dec 21, 2016

Replusz

I'd check that again...

4. Dec 21, 2016

Replusz

Yes, you can thus you are wrong :)

5. Dec 21, 2016

Logical Dog

Well, we had these questions for a level. But it doesent make sense to me.

Dont you mean if length of vector x is Y units, and vector X = (-2/3,b,16/7), thus find the value of b.

Otherwise how.. You must have been given a length for the vector, and then told to find the unknown component.? unless length of vector can also be zero.

6. Dec 21, 2016

Replusz

It does make sense, hwever its strange how it doesnt matter what a and c are :)

7. Dec 21, 2016

Logical Dog

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edit: got it. sorry, long time

Last edited: Dec 21, 2016
8. Dec 21, 2016

Replusz

Only if a=b=c=0 then the length l=sqrt(0)=0, yes. But this wasnt the question

9. Dec 21, 2016

cathal84

thanks for your reply, since it is possible to find the unknown variable, could you possibly point me in the right direction on how to do so?

10. Dec 21, 2016

Replusz

the length is sqrt(a^2+b^2+c^2)=sqrt(somehtin+b^2)
something is a number, like 2 or 5, or .... You miscalculated it though. It is a^2+c^2
Now a root is minimal if the thing inseide it is minmal. So b^2 + something is minimal. So b^2 is minimal
So now, what is b?

11. Dec 23, 2016

cathal84

still struggling with the question, i don't believe i have come across the concept of a minimal root and can't seem to find it in my text book, is it possible you could explain further? thanks

12. Dec 23, 2016

Ray Vickson

You are over-thinking the problem. If $b$ is allowed to be any real number whatsoever, what is the smallest possible value of $b^2$?

13. Dec 23, 2016

cathal84

Ah right, that makes a lot more senses haha thanks Ray and Replusz for your contribution.

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