How to find one vector of w in the Euclidean inner product inside a vector spaceif ⟨ , ⟩ symbolizes the Euclidean inner product inside the vector space R 3 , find one vector of w for which ⟨ w , v 1 ⟩ + ⟨ w , v 2 ⟩ = 2 ⟨ w , v 3 ⟩where v 1 = [ 1 , 0 , 1 ] , v 2 = [ 1 , 2 , 1 ] , v 3 = [ 1 , 3 , 10 ]How can I solve this?

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How to find one vector of w in the Euclidean inner product inside a vector spaceif   ⟨        ,     ⟩ symbolizes the Euclidean inner product inside the vector space             R        3  , find one vector of w for which  ⟨  w  ,      v    1    ⟩  +  ⟨  w  ,      v    2    ⟩  =  2  ⟨  w  ,      v    3    ⟩where       v    1    =  [  1  ,  0  ,  1  ]  ,      v    2    =  [  1  ,  2  ,  1  ]  ,      v    3    =  [  1  ,  3  ,  10  ]How can I solve this?

Claudia Woods · Accepted Answer

Use common logarithms and a decent calculator. Using a fairly mediocre one on your example, I get   99      log          10        ⁡  99  ≈  197.5678842652; subtracting   197 and raising   10 to the resulting power, I get   3.697296376497, so the number must be about   3.697296376497  ×      10          197      Added: More generally, for       a    b   calculate   b      log          10        ⁡  a, subtract the integer part, and raise   10 to the resulting power. If   n  =  ⌊  b      log          10        ⁡  a  ⌋, the integer part of   b      log          10        ⁡  a, your number is      10          b              log                  10                    ⁡      a      −      n        ×      10    n      ,and you’ll be able to read of the most significant digits from the lefthand end of       10          b              log                  10                    ⁡      a      −      n      .

Widersinnby7 · Accepted Answer

Let the unknown vector   w  =  [  x  ,  y  ,  z  ] where   (  x  ,  y  ,  z  ) are variables. Now, plug it into the given constraints:                                  ⟨        w        ,                  v          1                ⟩        +        ⟨        w        ,        v        2        ⟩        =        2        ⟨        w        ,                  v          3                ⟩                                          ⟨        [        x        ,        y        ,        z        ]        ,        [        1        ,        0        ,        1        ]        ⟩        +        ⟨        [        x        ,        y        ,        z        ]        ,        [        1        ,        2        ,        1        ]        ⟩        =        2        ⟨        [        x        ,        y        ,        z        ]        ,        [        1        ,        3        ,        10        ]        ⟩                                          (        x        +        z        )        +        (        x        +        2        y        +        z        )        =        2        (        x        +        3        y        +        10        z        )                                  2        x        +        2        y        +        2        z        =        2        x        +        6        y        +        20        z                                  2        y        +        2        z        =        6        y        +        20        z                                          −        4        y        =        18        z                                  y        =                              −            18            z                    4                    So,we have no constraints on x, and we know that   y  =            −      18      z        4  . Therefore,   w  =  [  x  ,  −  18  z      /    4  ,  z  ]  ,  ∀  x  ,  z  ∈      R

if (:,:) symbolizes the Euclidean inner product inside the vector space R3, find one vector of w for which (:w,v_1:)+(:w,v_2:)=2(:w,v_3:) where v_1=[1,0,1],v_2=[1,2,1],v_3=[1,3,10]

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