Let c and d be fixed vectors in R 3 . Find all vectors r such that r + r × d = c Attempt: Take the vector product with r : r × r + r × ( r × d ) = r × c ⇒ r × r + ( r ⋅ d ) r − ( r ⋅ r ) d = r × c ⇒ r × r = r × c ⇒ r × c = 0Hence any vector parallel to the vector c will work.

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Let c and d be fixed vectors in             R      3      . Find all vectors r such that       r    +      r    ×      d    =      c  Attempt: Take the vector product with       r  :       r    ×      r    +      r    ×  (      r    ×      d    )  =      r    ×      c    ⇒      r    ×      r    +  (      r    ⋅      d    )      r    −  (      r    ⋅      r    )      d    =      r    ×      c    ⇒      r    ×      r    =      r    ×      c    ⇒      r    ×      c    =  0Hence any vector parallel to the vector c will work.

Nezveda6q · Accepted Answer

You have   r  ×  d here, so to solve for r you need to calculate the dot product and the cross product with d instead of r.  (  r  ⋅  d  )  +  (  r  ×  d  )  ⋅  d  =  (  r  ⋅  d  )  +  0  =  (  r  ⋅  d  )  =  (  c  ⋅  d  )  r  ×  d  +  (  r  ×  d  )  ×  d  =  (  c  −  r  )  +  (  r  ⋅  d  )    d  −  (  d  ⋅  d  )     r  =  c  ×  dWhich allows to isolate r.  (  1  +  ‖  d      ‖    2    )    r  =  c  −  c  ×  d  +  (  r  ⋅  d  )    d  =  c  −  c  ×  d  +  (  c  ⋅  d  )    dTherefore  r  =                    c        +        (        c        ⋅        d        )                d        −        c        ×        d                    1        +        ‖        d                  ‖          2

Corinne Woods · Accepted Answer

In terms of vectors and matrices, the equation can be rewritten as  (  I  −  [      d        ]    ×    )      r    =      c    .By a change of orthonormal basis, we may assume that       d    =  (  0  ,  0  ,  a      )    T  . Therefore  [      d        ]    ×    =      (                            0                          −          a                          0                                      a                          0                          0                                      0                          0                          0                      )  and  (  I  −  [      d        ]    ×        )          −      1        =            (                                    1                                a                                0                                                −            a                                1                                0                                                0                                0                                1                              )              −      1        =      (                                        1                          1              +                              a                2                                                                                        −              a                                      1              +                              a                2                                                              0                                                  a                          1              +                              a                2                                                                          1                          1              +                              a                2                                                              0                                      0                          0                          1                      )    =      1          1      +      ‖              d                    ‖        2                  (    I    +          d                      d            T        +    [          d              ]      ×        )    .It follows that      r    =  (  I  −  [      d        ]    ×        )          −      1            c    =      1          1      +      ‖              d                    ‖        2                  (          c        +    (          d        ⋅          c        )          d        +          d        ×          c        )    .

Let c and d be fixed vectors in RR^3. Find all vectors r such that r+r xx d=c. Attempt: Take the vector product with r: r xx r+r xx (r xx d)=r xx c => r xx r+(r * d)r−(r*r)d=r xx c => r xx r=r xx c => r xx c=0. Hence any vector parallel to the vector c will work.

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