Three vectors, V 1 , V 2 , V 3 are in the R 2 plane where V 1 + V 2 + V 3 = 0 → and the magnitudes of these vectors are the same. Show that the angle between any two of these vectors is 120 degrees.My try: V 1 + V 2 + V 3 = 0 → , so V 1 + V 2 = − V 3 . This would mean that ( V 1 + V 2 ) 2 = V 3 2 which is | V 1 | + | V 2 | + 2 V 1 V 2 = | V 3 | . But since the magnitude is the same, we get 2 | V 1 | + 2 V 1 V 2 = | V 1 | which is 2 V 1 V 2 = − | V 1 | The formula for the angle is cos ⁡ ( a ) = V 1 ∗ V 2 | V 1 | ∗ | V 2 | = − 0.5 ∗ | V 1 | | V 1 | 2 = − 0.5 | V 1 | I know that the solution would be a triangle and that cos ⁡ ( a ) = − 0.5But that would mean that my angle is dependent on V 1 which isn't the case. So what am I doing wrong?

Question

Three vectors,       V    1    ,      V    2    ,      V    3   are in the             R        2   plane where       V    1    +      V    2    +      V    3    =            0      →       and the magnitudes of these vectors are the same. Show that the angle between any two of these vectors is 120 degrees.My try:      V    1    +      V    2    +      V    3    =            0      →      , so       V    1    +      V    2    =  −      V    3  . This would mean that   (      V    1    +      V    2        )    2    =      V    3    2   which is       |        V    1        |    +      |        V    2        |    +  2      V    1        V    2    =      |        V    3        |  . But since the magnitude is the same, we get   2      |        V    1        |    +  2      V    1        V    2    =      |        V    1        |   which is   2      V    1        V    2    =  −      |        V    1        |  The formula for the angle is   cos  ⁡  (  a  )  =                    V        1            ∗              V        2                            |                    V        1                    |            ∗              |                    V        2                    |              =  −  0.5  ∗                    |                    V        1                    |                            |                    V        1                              |                2              =            −      0.5                      |            V      1              |            I know that the solution would be a triangle and that   cos  ⁡  (  a  )  =  −  0.5But that would mean that my angle is dependent on       V    1   which isn&#039;t the case. So what am I doing wrong?

Jake Landry · Accepted Answer

V    3    =  −      V    1    −      V    2   implies that   ‖      V    3        ‖    2    =  ⟨      V    1    +      V    2    ,      V    1    +      V    2    ⟩  =  ‖      V    1        ‖    2    +  ‖      V    2        ‖    2    +  2  ⟨      V    1    ,      V    2    ⟩  =  ‖  V      ‖    2   where   ‖      V    1    ‖  =  ‖      V    2    ‖  =  ‖      v    3    ‖  =  ‖  V  ‖This implies that   2  ‖  V      ‖    2    c  o  s  (      V    1    ,      V    2    )  =  −  ‖  V      ‖    2

Landon Wolf · Accepted Answer

You cannot &quot;square&quot; a vector: the statement &quot;  (      V    1    +      V    2        )    2    =      V    3    2  &quot; makes no sense.I sense that you are trying to use the dot product. In that case, you will instead obtain:      |        V    3              |        2    =      V    3    ⋅      V    3    =  (      V    1    +      V    2    )  ⋅  (      V    1    +      V    2    )  =      |        V    1              |        2    +  2      V    1    ⋅      V    2    +      |        V    2              |        2  Now we continue with your method. We have from      |        V    1              |        2    =  −  2      V    1    ⋅      V    2      ⟹    cos  ⁡  (  a  )  =                    V        1            ⋅              V        2                            |                    V        1                    |                    |                    V        2                    |              =                    V        1            ⋅              V        2                            |                    V        1                              |                2              =  −      1    2

Three vectors, V_1,V_2,V_3 are in the RR^2 plane where V_1+V_2+V_3=vec(0) and the magnitudes of these vectors are the same. Show that the angle between any two of these vectors is 120 degrees.

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