Find the area of a triangle with vertices ( 0 , 1 , 1 ) , ( − 1 , − 1 , 2 ) , ( 2 , 3 , 1 )

Question

Find the area of a triangle with vertices   (  0  ,  1  ,  1  )  ,  (  −  1  ,  −  1  ,  2  )  ,  (  2  ,  3  ,  1  )

lutzantsca885 · Accepted Answer

Step 1Let us put                    A                            :=        (        −        1        ,        −        1        ,        2        )                            B                            :=        (        0        ,        1        ,        1        )                            C                            :=        (        2        ,        3        ,        1        )        .             .Then the vectors             A      B        →   and             A      C        →   are given byAB→=i^+2j^−k^AC→=3i^+4j^−k^. .So                                          A            B                    →                ×                              A            C                    →                                    =                  |                                                                                          i                    ^                                                                                                                    j                    ^                                                                                                                    k                    ^                                                                                                      1                                            2                                            −                1                                                                    3                                            4                                            −                1                                              |                                                  =        2                              i            ^                          −        2                              j            ^                          −        2                              k            ^                              Therefore our required area is      1    2        |                  A        B            →        ×                  A        C            →        |    =      1    2        (    2          )      2        +    (    −    2          )      2        +    (    −    2          )      2        =      1    2        12    =      1    2    2      3    =      3    , ,as required.This method is much shorter than the Heron&#039;s formula.

Kazeljkaml5n9y · Accepted Answer

Step 1You can express two sides as free vectors   x  =  (  −  1  ,  −  2  ,  1  ) ,   y  =  (  2  ,  2  ,  0  ). Then, the area A can also be expressed with dot products (see here) as  A  =      1    2        (    x    x    )    (    y    y    )    −    (    x    y          )      2          =      1    2        6    ⋅    8    −    (    −    6          )      2        =      1    2        12    =      3  I guess this is the fastest method.

Find the area of a triangle with vertices ( 0 , 1 , 1 ) , ( −<

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