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This two question are from an exam. They are not difficult but I think maybe the answer is not in the choices given in the test. I want to know if I am right or not

  1. Let vectors $a = (2,3,1) , b=(-2,0,3)$ be two sides of a Parallelogram. What is the length of its longest diagonal?

1.$\sqrt{26}$ ||| 2.$4$ ||| 3.$5$ ||| 4. $\sqrt{21}$

My solution is we know that the diameters are a +b and a-b so $a+b = (0 , 3,4)$ and $a-b= (4 , 3 , -2)$ So length of a diagonal is 5 and the other is $\sqrt{29}$ which is obviously greater than 5 so the answer is $\sqrt{29}$ which is not in the answers. Is my solution right?

The second question is about cauchy-schwartz inequality in vectors $|a||b|\geq |a.b|$

2.Let $6x-y+4z =12$ and $9x^2 + y^2 + 4z^2$ is minimum. Find $9x +3y +3z$ .

  1. $4$ ||| 2.$6$ ||| 3.$8$ ||| 4.$10$

My solution is $|(3x,y,2z).(2,-1,2)|=6x-y+4z=12 \leq |(3x,y,2z)||(2,-1,2)|$ so because of it says "$9x^2 + y^2 + 4z^2$ is minimum" so $|(3x,y,2z)|$ = 4. I also assumed that I must multiply this answer by $|(3,3,3/2)| = 9/2$ so I could get $9x +3y +2z$ and it is 18 which is not in the answers. Is my solution right? if not, what is the right solution?

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  • $\begingroup$ what do you mean with diameter ? $\endgroup$ Dec 10, 2017 at 15:28
  • $\begingroup$ @Dr.SonnhardGraubner I mean diagonal. I am not native English and didn't know that in English, Diameter and diagonal are different. We have the same word for them in my language. (I mean, for example, in ABCD, AC and BD are diagonal) $\endgroup$
    – titansarus
    Dec 10, 2017 at 15:32
  • $\begingroup$ @Dr.SonnhardGraubner The concept of "diameter" is well defined even though OP doesn't specify it. In this case, it's obvious that this means the longest diagonal. $\endgroup$ Dec 10, 2017 at 15:34
  • $\begingroup$ @titansarus If you are ok, you can set as solved. Thanks! $\endgroup$
    – user
    Dec 13, 2017 at 7:29

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I agree with both your solution! Your calculation and reasoning seems completely correct.

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  • $\begingroup$ This does not provide an answer to the question. To critique or request clarification from an author, leave a comment below their post. - From Review $\endgroup$ Dec 10, 2017 at 16:23
  • $\begingroup$ His question is precisely: " is my solution right. if not, what is the right solution?". My answer is that that his solution is right, $\endgroup$
    – user
    Dec 10, 2017 at 16:24
  • $\begingroup$ @infinitylord How exactly does this not answer the OP’s question? $\endgroup$
    – amd
    Dec 10, 2017 at 18:30

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