Maximum value of | a → − 2 b → | 2 + | b → − 2 c → | 2 + | c → − 2 a → | 2 I have a question the goes, if | a → | = | b → | = 1 and | c → | = 2, then what is the maximum value of : | a → − 2 b → | 2 + | b → − 2 c → | 2 + | c → − 2 a → | 2 ?Here's what I tried: | a → − 2 b → | 2 + | b → − 2 c → | 2 + | c → − 2 a → | 2 = 30 − 4 cos ⁡ θ 1 − 8 cos ⁡ θ 2 − 8 cos ⁡ θ 3 where θ 1 = angle between a → and b → , θ 2 = angle between b → and c → , θ 3 = angle between c → and a → and for maximum value, I put cos ⁡ θ 1 = cos ⁡ θ 2 = cos ⁡ θ 3 = − 1, and got the maximum value of 50 but the answer says 42. Where did I go wrong and what would be the correct approach? Was I wrong in putting cos ⁡ θ 1 = cos ⁡ θ 2 = cos ⁡ θ 3 = − 1? Why?

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Maximum value of       |        a    →    −  2      b    →              |        2    +      |        b    →    −  2      c    →              |        2    +      |        c    →    −  2      a    →              |        2  I have a question the goes, if       |        a    →        |    =      |        b    →        |    =  1 and       |        c    →        |    =  2, then what is the maximum value of :      |        a    →    −  2      b    →              |        2    +      |        b    →    −  2      c    →              |        2    +      |        c    →    −  2      a    →              |        2  ?Here&#039;s what I tried:      |        a    →    −  2      b    →              |        2    +      |        b    →    −  2      c    →              |        2    +      |        c    →    −  2      a    →              |        2    =  30  −  4  cos  ⁡      θ    1    −  8  cos  ⁡      θ    2    −  8  cos  ⁡      θ    3  where       θ    1    =  angle between       a    →   and       b    →   ,       θ    2    = angle between       b    →   and       c    →  ,       θ    3    = angle between       c    →   and       a    →  and for maximum value, I put   cos  ⁡      θ    1    =  cos  ⁡      θ    2    =  cos  ⁡      θ    3    =  −  1, and got the maximum value of 50 but the answer says 42. Where did I go wrong and what would be the correct approach? Was I wrong in putting   cos  ⁡      θ    1    =  cos  ⁡      θ    2    =  cos  ⁡      θ    3    =  −  1? Why?

rcampas4i · Accepted Answer

By putting all   cos  ⁡      θ    i    =  −  1 you imply that all the vectors make an angle   π with each other !start off with                    |                              a          →                    +                        b          →                    +                        c          →                            |              2    ≥  0to get  −  ∑            a      →        ⋅            b      →        ≤  3therefore  ∑                    |                              a          →                    −                                    2            b                    →                            |              2    =  30  −  4  ∑            a      →        ⋅            b      →        ≤  42

I have a question the goes, if |vec(a)|=|vec(b)|=1 and |vec(c)|=2, then what is the maximum value of : |vec(a)−2vec(b)|^2+|vec(b)−2vec(c)|^2+|vec(c)−2vec(a)|^2 ?

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