For what t will the following vector be an orthogonal basis? u 1 = ( 1 , t , t ) u 2 = ( 2 t , t + 1 , 2 t − 1 ) u 3 = ( 2 − 2 t , t − 1 , 1 ) Till now I have tried using the Gram-Schmidt process but did not really reach anywhere. Can you please provide a hint or some theory that may help me get the solution for this question?

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For what t will the following vector be an orthogonal basis?                              u          1                                    =        (        1        ,        t        ,        t        )                                      u          2                                    =        (        2        t        ,        t        +        1        ,        2        t        −        1        )                                      u          3                                    =        (        2        −        2        t        ,        t        −        1        ,        1        )            Till now I have tried using the Gram-Schmidt process but did not really reach anywhere. Can you please provide a hint or some theory that may help me get the solution for this question?

merneh7 · Accepted Answer

You can easily see that whatever be the value of t, we have             u      3        =      2          u      1        −            u      2      . Therefore,   span  {      u    1    ,      u    2    ,      u    3    }  =  span  {      u    1    ,      u    2    } and we only need to perform orthogonalization for       u    1    ,      u    2  Using the Gram-Schmidt process, we have   B  =  {      v    1    ,      v    2    }, where:            v      1        =            u      1        =  (  1  ,  t  ,  t  )                    v        2              =                  u        2              −                  ⟨                              u            1                          ,                              u            2                          ⟩                    ⟨                              u            1                          ,                              u            1                          ⟩              ⋅                  u        1              =    (    2    t    ,    t    +    1    ,    2    t    −    1    )    −                  [                            3                  t          2                +        2        t                    1        +        2                  t          2                                    ]              (    1    ,    t    ,    t    )          =                  (                            4                  t          3                −        3                  t          2                            1        +        2                  t          2                      ,                  −                  t          3                +        t        +        1                    1        +        2                  t          2                      ,                            t          3                −        4                  t          2                +        2        t        −        1                    1        +        2                  t          2                                    )

For what t will the following vector be an orthogonal basis? u_1=(1,t,t) u_2=(2t,t+1,2t−1) u_3=(2−2t,t−1,1)

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