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Solutions to cross product, a x u = b

  1. Apr 8, 2013 #1
    Studying outer product spaces at the moment and thought I'd quickly recap on the cross product when I stumbled across this problem which has me fairly stumped!

    If a,b∈R^3 with a≠0 show that the equation a x u = b has a solution if and only if a.b = 0 and fi nd all the solutions in this case.

    The answer is,

    -((a x b)/|a|^2) + λ a , where λ is a real parameter.

    The first part is trivial, but I have no idea how to get to the solution set. Could anybody shed any light on this matter? I would be very grateful.
  2. jcsd
  3. Apr 8, 2013 #2


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    Staff: Mentor

    Can you show that a x [solutions] = b (for all λ)? Can you show that the set of solutions is one-dimensional?
    It is possible to derive the answer if you work out the individual components of the equation.
  4. Apr 10, 2013 #3
    First take the cross product of the equation with a:
    [itex]a \times (a \times u = b)[/itex]

    The vector triple product gives

    [itex](a \cdot u) a -a^2 u = a \times b [/itex]

    With a little algebra you can then get

    [itex]u =-\frac{ a \times b}{a^2} + \frac{a\cdot u}{a^2} a[/itex]
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