Trying to prove:Show that there are no vectors u and v such that ‖ u ‖ = 1, ‖ v ‖ = 2, and u ⋅ v = 3Don't know where to go from here: ‖ u ‖ = 2 ⟹ ‖ u ‖ 2 = 4 ⟹ u ⋅ u = 1 ‖ v ‖ = 1 ⟹ ‖ v ‖ 2 = 1 ⟹ v ⋅ v = 4Not sure if this is the right direction to take, but we have: u ⋅ v = v ⋅ v − u ⋅ u = 3 = ( v + u ) ⋅ ( v − u )

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Trying to prove:Show that there are no vectors u and v such that   ‖  u  ‖  =  1,   ‖  v  ‖  =  2, and   u  ⋅  v  =  3Don&#039;t know where to go from here:  ‖  u  ‖  =  2  ⟹  ‖  u      ‖    2    =  4  ⟹  u  ⋅  u  =  1  ‖  v  ‖  =  1  ⟹  ‖  v      ‖    2    =  1  ⟹  v  ⋅  v  =  4Not sure if this is the right direction to take, but we have:                    u        ⋅        v        =        v        ⋅        v        −        u        ⋅        u        =        3                            =        (        v        +        u        )        ⋅        (        v        −        u        )

Santiago Collier · Accepted Answer

u  ⋅  v  =  ‖  u  ‖  ‖  v  ‖  cos  ⁡  θSo  3  =  (  1  )  (  2  )  cos  ⁡  θwhich implies  cos  ⁡  θ  =            3      2      which is not possible.

easternerjx · Accepted Answer

Another way to realize the answer:                    ‖        u        +        v                  ‖                      2                                              =        ⟨        u        +        v        ,        u        +        v        ⟩                                                        =        ⟨        u        ,        u        ⟩        +        2        ⟨        u        ,        v        ⟩        +        ⟨        v        ,        v        ⟩                                                        =        ‖        u                  ‖                      2                          +        2        ⟨        u        ,        v        ⟩        +        ‖        v                  ‖                      2                                                                          =                  1                      2                          +        2        ×        3        +                  2                      2                                                                          =        1        +        6        +        4                                                        &amp;gt;        (        1        +        2                  )                      2                                                                          =        (        ‖        u        ‖        +        ‖        v        ‖                  )                      2                              which violates the triangle inequality.

Trying to prove: Show that there are no vectors u and v such that ∥u∥=1, ∥v∥=2, and u*v=3.

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