For f ( t ) = ( 1 1 - t , - 1 1 + t 2 ) what is the distance between f(2) and f(5)?

Question

For             f              (        t        )              =          (              1                  1          -          t                    ,      -              1                  1          +                      t            2                              )       what is the distance between f(2) and f(5)?

SuefsSeeltHeRn8 · Accepted Answer

Step 1Find the points first:            f              (        2        )              =          (              1                  1          -          2                    ,      -              1                  1          +                      2            2                              )        =          (      -      1      ,      -              1        5            )                  f              (        5        )              =          (              1                  1          -          5                    ,      -              1                  1          +                      5            2                              )        =          (      -              1        4            ,      -              1        26            )      The distance between the points             (              x        1            ,              y        1            )       and             (              x        2            ,              y        2            )       is calculated through       d    =                                        (                          y              2                        -                          y              1                        )                    2                +                              (                          x              2                        -                          x              1                        )                    2                          d    =                                        (            -                          1              26                        -                          (              -                              1                5                            )                        )                    2                +                              (            -                          1              4                        -                          (              -              1              )                        )                    2                          d    =                                        (                          21              130                        )                    2                +                              (                          3              4                        )                    2                    Step 2You can go ahead and make this a decimal answer, but you can also do some nice simplification if you feel up to it.      d    =                                        (                                          3                ⋅                7                                            2                ⋅                5                ⋅                13                                      )                    2                +                              (                          3                              2                2                                      )                    2                          d    =                                                      3              2                        ⋅                          7              2                                                          2              2                        ⋅                          5              2                        ⋅                          13              2                                      +                              3            2                                2            4                                    d    =                                                      2              2                        ⋅                          3              2                        ⋅                          7              2                                                          2              4                        ⋅                          5              2                        ⋅                          13              2                                      +                                            3              2                        ⋅                          5              2                        ⋅                          13              2                                                          2              4                        ⋅                          5              2                        ⋅                          13              2                                                d    =                                                      3              2                                      (                              2                2                            ⋅                              7                2                            +                              5                2                            ⋅                              13                2                            )                                                          2              4                        ⋅                          5              2                        ⋅                          13              2                                                d    =          3                        2          2                ⋅        5        ⋅        13                                                  (            2            ⋅            7            )                    2                +                              (            5            ⋅            13            )                    2                          d    =                  3                  4421                            260              ≈    0.767199

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