Use vectors to show that A(3,0,2), B(4,3,0), and C(8,1,-1) are vertices of a right triangle. At which vertex is the right angle?

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Isaias Archer · Accepted Answer

Step 1I computed the following vectors:                    A        B            →        =  (  4  ,  3  ,  0  )  −  (  3  ,  0  ,  2  )  =  (  1  ,  3  ,  −  2  )  ,                    A        C            →        =  (  5  ,  1  ,  −  3  )  ,                    B        C            →        =  (  4  ,  −  2  ,  −  1  )  .Now you see that                     A        B            →        ⋅                    B        C            →        =  (  1  ,  3  ,  −  2  )  ⋅  (  4  ,  −  2  ,  −  1  )  =  1  ×  4  +  3  ×  (  −  2  )  +  (  −  2  )  ×  (  −  1  )  =  0  .Step 2Therefore   A  B  ⊥  B  C and the right angle is at B.

polissemkt · Accepted Answer

Step 1Angle B: between vectors             B      A        →    =  (  −  1  ,  −  3  ,  2  ) and             B      C        →    =  (  4  ,  −  2  ,  −  1  ) (heads directed away from the (origin) vertex B) is given as   ∠  B  =      cos          −      1        ⁡      (                  (        −        1        )        (        4        )        +        (        −        3        )        (        −        2        )        +        (        2        )        (        −        1        )                              (                      (            −            1                          )              2                        +            (            −            3                          )              2                        +            (            2                          )              2                                )                          (                      (            4                          )              2                        +            (            −            2                          )              2                        +            (            −            1                          )              2                                )                      )    =      cos          −      1        ⁡      (          0              294              )    =      cos          −      1        ⁡      (    0    )    =      90    o  Hence,   Δ  A  B  C is a right triangle with right angle at the vertex B (  ∠  B  =      90    o  )

Use vectors to show that A(3,0,2), B(4,3,0), and C(8,1,-1) are vertices of a right triangle. At which vertex is the right angle?

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