Given v 1 → × v 2 → and v 1 → × v 3 → , find v 2 → × v 3 → Is it possible to get the cross product of two vectors v 2 → and v 3 → , given the result of each cross product with a common vector v 1 → ?e.g. v i → ∈ R + 2 ‖ v 1 → × v 2 → ‖ = 0.10007117 ‖ v 1 → × v 3 → ‖ = 0.05423194 find ‖ v 2 → × v 3 → ‖ . which is 0.01196676, since i randomly generated 3 vectors: v 1 = ( 0.97986514 , 0.40872297 ) v 2 = ( 0.69637269 , 0.39259962 ) v 3 = ( 0.2602131 , 0.16388685 )Is there any way to get to the answer without knowing the explicit vectors?

Question

Given                     v        1            →        ×                    v        2            →       and                     v        1            →        ×                    v        3            →      , find                     v        2            →        ×                    v        3            →      Is it possible to get the cross product of two vectors                     v        2            →       and                     v        3            →      , given the result of each cross product with a common vector                     v        1            →      ?e.g.                    v        i            →        ∈            R        +    2      ‖                    v        1            →        ×                    v        2            →        ‖  =  0.10007117    ‖                    v        1            →        ×                    v        3            →        ‖  =  0.05423194  find   ‖                    v        2            →        ×                    v        3            →        ‖ . which is   0.01196676, since i randomly generated 3 vectors:      v    1    =  (  0.97986514  ,  0.40872297  )        v    2    =  (  0.69637269  ,  0.39259962  )        v    3    =  (  0.2602131  ,  0.16388685  )Is there any way to get to the answer without knowing the explicit vectors?

Olive Avila · Accepted Answer

No. For example,                     v        1            →        =  (  0  ,  1  )The following to set of                     v        2            →        ,                    v        3            →       give the same                     v        1            →        ×                    v        2            →       and                     v        1            →        ×                    v        3            →      , yet their                     v        2            →        ×                    v        3            →       are different.                    v        2            →        =  (  1  ,  1  )  ,                    v        3            →        =  (  2  ,  1  )                    v        2            →        =  (  1  ,  10  )  ,                    v        3            →        =  (  2  ,  10  )

FoomyBiamy5 · Accepted Answer

Here is a counter-example       {                                                                      v                1                            →                                =          (          a          ,          b          )                                                                                v                2                            →                                =          (                      α            b                    ,                                    2              α                        a                    )                                                                                v                3                            →                                =          (                                    α              β              +              a              b              γ                                      b              α                                ,                                    2              α              β              +              a              b              γ                                      a              α                                )                         then       {                            ‖                                                    v                1                            →                                ×                                                    v                2                            →                                ‖          =          α                                      ‖                                                    v                1                            →                                ×                                                    v                3                            →                                ‖          =          β                                      ‖                                                    v                2                            →                                ×                                                    v                3                            →                                ‖          =          γ                        with      a  ,  b  ,  α  ,  β  ,  γ  &amp;gt;  0Since all three numbers   α  ,  β  ,  γ can be chosen independently, it shows that the problem is not solvable in the general case.

Is it possible to get the cross product of two vectors vec(v_2) and vec(v_3), given the result of each cross product with a common vector vec(v_1)?

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