Suppose I have 4 unit vectors in 3D and I know all the 4 C 2 = 6 angles between them. These angles provide the complete description of this group of vectors. Now, I want to add anther unit vector to the mix. How many additional angles do I need to uniquely identify this new vector?

Question

Suppose I have 4 unit vectors in 3D and I know all the           4        C    2    =  6 angles between them. These angles provide the complete description of this group of vectors. Now, I want to add anther unit vector to the mix. How many additional angles do I need to uniquely identify this new vector?

noruegezajl00y · Accepted Answer

Step 1You need to pick two vectors       V    1   and       V    2   from the four vectors that you have, and specify angles of the added vector (the fifth vector) from these two vectors. There can be two vectors making       θ    1   with       V    1   and       θ    2   with       V    2  . So you just need one more angle with a third vector       V    3   to uniquely specify your added vector, but keep in mind that this third angle can take only two specific values.As an explicit numerical example, suppose      V    1    =  (  1  ,  0  ,  0  )      V    2    =  (  0  ,  1  ,  1  )      /        2        V    3    =  (  2  ,  0  ,  −  1  )      /        5        V    4    =  (  1  ,  1  ,  1  )      /        3  Now suppose I have a fifth unit vector       V    5    =  (  1  ,  2  ,  3  )      /        14  That I want to uniquely identify using angles from the given vectors.We note that      V    1    ⋅      V    5    =            1              14              =  cos  ⁡      θ    1  and      V    2    ⋅      V    5    =            5              28              =  cos  ⁡      θ    2  and      V    3    ⋅      V    5    =                    −        1                    70              =  cos  ⁡      θ    3  And now we want to go from the       θ    1    ,      θ    2    ,      θ    3   to the vector       V    5   .So we&#039;ll let       V    5    =  a      V    1    +  b      V    2    +  c      V    3   then using the dot product between       V    5   and       V    1    ,      V    2    ,      V    3   we&#039;ll have the linear system in the cosines of the angles between V5 and       V    1    ,      V    2    ,      V    3   as follows      [                                        V            1                    ⋅                      V            5                                                            V            2                    ⋅                      V            5                                                            V            3                    ⋅                      V            5                                ]    =      [                            1                                              V            1                    ⋅                      V            2                                                        V            1                    ⋅                      V            3                                                            V            2                    ⋅                      V            1                                            1                                              V            2                    ⋅                      V            3                                                            V            3                    ⋅                      V            1                                                        V            3                    ⋅                      V            2                                            1                      ]        [                            a                                      b                                      c                      ]  Substituting the values in the above matrix equation we arrive at            [                                                                  1                                  14                                                                                                                        5                                  28                                                                                                                                          −                  1                                                  70                                                                        ]        =          [                                    1                                          0                                                                        2                                  5                                                                                          0                                          1                                          −                                          1                                  10                                                                                                                        2                                  5                                                                                    −                                          1                                  10                                                                                    1                              ]              [                                    a                                                b                                                c                              ]      Solving this system numerically, gives us  a  =  0.801784  ,  b  =  0.755929  ,  c  =  −  0.59761Now using      V    5    =  a      V    1    +  b      V    2    +  c      V    3  gives us                              V          5                                    =        (        0.267261        ,        0.534522        ,        0.801784        )                                          =        (        0.267261        )        (        1        ,        2        ,        3        )                                          =        (        1        ,        2        ,        3        )                  /                          14

Suppose I have 4 unit vectors in 3D and I know all the 4 </msup> C

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