# Resolving Cross Product Ambiguity Algebraically

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?

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