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Say we have an orthonormal basis in $\mathbb R^3$ $\{A, B, C\}$. Then when taking the cross product of any 2 of these, we know it is equal to either the third basis element, or $-1$ times that element.

Say we are given $A \times B = C$, and we want to know whether $A \times C = B$ or $A \times C = -B$.

Via the right-hand rule, I can see that it should be $-B$, but how can I show this algebraically, using only what is given here?

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You can use the vector triple product identity: $$x\times(y\times z)=(x\cdot z)y-(x\cdot y)z$$ Then$$A\times C=A\times(A\times B)=(A\cdot B)A-(A\cdot A)B=0A-1B=-B$$

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