Does the following cross product identity always work?

[em370]

Mod note: Reproduced contents of image with broken link:
i = j x k
j = k x i
k = i x j
Wikipedia says this about the standard basis vectors. Does this work for all (i.e, non basis) vectors? For example, if you know A = B X C does that mean C = A X B and B = C X A?

 

[Fightfish]

Nope. Let's start with $A = B \times C$ and see what $A \times B$ gives us. Using the vector triple product identity, we have $A \times B = (B\times C) \times B = C (B\cdot B) - B (B \cdot C)$. So, $A \times B = C$ only if $B$ and $C$ are orthogonal (i.e. their dot product is zero) - which is true for the standard basis vectors, but not true in general.

 

[fresh_42]

Have you tried $A=0$?