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For example, $a \times b = c$

If you only know $a$ and $c$, what method can you use to find $b$?

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It's a set of linear equations. So, you use linear algebra to solve. – Raskolnikov Apr 12 '11 at 19:13
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$b$ is not uniquely determined by the knowledge of $a$ and $c$. Do you want to add some additional conditions on $b$ or are you interested in all possible solutions? – Fabian Apr 12 '11 at 19:13

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As Fabian wrote, $b$ is not uniquely determined by $a$ and $c$. Moreover, there is no solution unless $a$ and $c$ are orthogonal. If $a$ and $c$ are orthogonal, then the solutions are $(c \times a)/(a\, . a) + t a$ for arbitrary scalars $t$.

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To spell this out in words I understand, $b$ lies in the plane orthogonal to $c$ (as must $a$). The component of $b$ orthogonal to $a$ is $|c|/|a|$ in the direction consistent with the right-hand rule. – Henry Apr 13 '11 at 0:12
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The name "product" for the cross product is unfortunate. It really should not be thought of as a product in the ordinary sense; for example, it is not even associative. Thus one should not expect it to have properties analogous to the properties of ordinary multiplication.

What the cross product really is is a Lie bracket.

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