# Solutions to cross product, a x u = b

1. Apr 8, 2013

### xtrap0lation

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.

-((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. Apr 8, 2013

### 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.

3. Apr 10, 2013

### the_wolfman

First take the cross product of the equation with a:
$a \times (a \times u = b)$

The vector triple product gives

$(a \cdot u) a -a^2 u = a \times b$

With a little algebra you can then get

$u =-\frac{ a \times b}{a^2} + \frac{a\cdot u}{a^2} a$