Calculate angle between two vectors, given their rotation w.r.t. a third vector.I have three vectors, a → , b → , and c → in n-dimensional space. I know the coordinates of all three vectors and their dot products. Both a → and b → are rotated away from c → by an angle α, in their own respective directions, obtaining a → ′ and b → ′ . What is the angle between a → ′ and b → ′ ?

Question

Calculate angle between two vectors, given their rotation w.r.t. a third vector.I have three vectors,             a      →        ,            b      →        , and             c      →        in n-dimensional space. I know the coordinates of all three vectors and their dot products. Both             a      →       and             b      →       are rotated away from             c      →       by an angle   α, in their own respective directions, obtaining                     a        →              ′   and                     b        →              ′   . What is the angle between                     a        →              ′   and                     b        →              ′   ?

sniokd · Accepted Answer

Step 1Assuming that a,b,c have norm 1, we have                              a          ′                                    =        2        (        a        ⋅        c        )        a        −        c                                      b          ′                                    =        2        (        b        ⋅        c        )        b        −        c            Indeed, this implies       a    ′    ⋅  a  =  a  ⋅  c and       a    ′    ⋅  c  =  2  (  a  ⋅  c      )    2    −  1 which is the cosine of the double angle. Alternatively, it is obvious geometrically that                     a        ′            +      c        2   is the orthogonal projection of c on a, that is to say   (  a  ⋅  c  )  aHence      a    ′    ⋅      b    ′    =  4  (  a  ⋅  c  )  (  b  ⋅  c  )  (  a  ⋅  b  )  −  2  (  a  ⋅  c      )    2    −  2  (  b  ⋅  c      )    2    +  1

Calculate angle between two vectors, given their rotation w.r.t. a third vector. I have three vecto

Answered question

Answer & Explanation

New Questions in High school geometry