I'm reading a textbook at the moment that provides the following linear equation, α v + v × a = b ,and asks to solve for v . The form of v is given as v = α 2 b − α ( b × a ) + ( a ⋅ b ) a α ( α 2 + | a | 2 ) .It's easy enough to verify that this is the correct solution. However, I can't figure out how I'd solve for v if given just the original equation.Are there any general approaches to solving this kind of equation systematically?Edit: a , b and v are all vectors, whereas α is a scalar such that α ≠ 0.

Question

I&#039;m reading a textbook at the moment that provides the following linear equation,  α      v    +      v    ×      a    =      b    ,and asks to solve for       v  . The form of       v   is given as      v    =                    α        2                    b            −      α      (              b            ×              a            )      +      (              a            ⋅              b            )              a                    α      (              α        2            +      |              a                    |        2            )        .It&#039;s easy enough to verify that this is the correct solution. However, I can&#039;t figure out how I&#039;d solve for       v   if given just the original equation.Are there any general approaches to solving this kind of equation systematically?Edit:       a  ,      b   and       v   are all vectors, whereas   α is a scalar such that   α  ≠  0.

Colin Moran · Accepted Answer

Taking cross product with       a   on both sides, we get,                          α                  v                +                  v                ×                  a                =                  b                                            ⟹                            α        (                  v                ×                  a                )        +        (                  v                ×                  a                )        ×                  a                =                  b                ×                  a                                            ⟹                            α        (                  b                −        α                  v                )        +        (                  v                ⋅                  a                )                  a                −                  |                a                              |                    2                          v                =                  b                ×                  a                                            ⟹                            α                  b                −                  α          2                          v                +                              1            α                          (                  b                ⋅                  a                )                  a                −                  |                a                              |                    2                          v                =                  b                ×                  a                                                                (                          Using         α        (                  v                ⋅                  a                )        =                  b                ⋅                  a                                      )                              Now solve for       v   directly.