Prove that u → and v → are orthogonal vectorsThe vectors u → and v → are given in terms of the basis vectors a → , b → and c → as follows: u → = 3 a → + 3 b → − c → v → = a → + 2 b → + 3 c → I've tried u → . v → to see if their dot product equals to 0, but it does not. Am I missing something?It was given that a → , b → , and c → form a basis in R 3 .It was also given that: | a → | = 1 , | b → | = 2 , | c → | = 3

Question

Prove that             u      →        and             v      →        are orthogonal vectorsThe vectors             u      →       and             v      →       are given in terms of the basis vectors             a      →       ,             b      →        and             c      →        as follows:            u      →        =  3            a      →        +  3            b      →        −            c      →                  v      →        =            a      →        +  2            b      →        +  3            c      →      I&#039;ve tried             u      →        .            v      →        to see if their dot product equals to 0, but it does not. Am I missing something?It was given that             a      →        ,            b      →        , and             c      →        form a basis in       R    3  .It was also given that:      |              a      →            |    =  1  ,      |              b      →            |    =  2  ,      |              c      →            |    =  3

Steppkelk · Accepted Answer

With these changes,  ⟨  u  ,  v  ⟩  =  ⟨  3  a  +  3  b  −  c  ,  a  +  2  b  +  3  c  ⟩                    =                    !              ⟨  3  a  ,  a  ⟩  +  ⟨  3  b  ,  2  b  ⟩  +  ⟨  −  c  ,  3  c  ⟩  =  3  ⟨  a  ,  a  ⟩  +  6  ⟨  b  ,  b  ⟩  −  3  ⟨  c  ,  c  ⟩  =  3  ⋅      1    2    +  6  ⋅      2    2    −  3  ⋅      3    2    =  0But you still need that the basis is orthogonal.

makaunawal5 · Accepted Answer

If you expand   u  ⋅  v you find   9  a  ⋅  b  +  8  a  ⋅  c  +  7  b  ⋅  cThere is no reason for this to be zero in general, unless we also know that a,b,c are orthogonal. So my guess is that in the instructions for the problem there was the information that a,b,c are orthogonal.