Why are cross products in different coordinates different?

Question

For example: In coordinate (1): $[e_1, e_2, e_3]$. Let $v_1 = [1, 0, 0]$, $v_2 = [0, 1, 0]$, $v_1 \times v_2 = [0, 0, 1]$ in (1).

In coordinate (2): $[f_1, f_2, f_3] = [e_1, e_2, e_3]/2$. So $v_1 = [2, 0, 0]$, $v_2 = [0, 2, 0]$. $v_1 \times v_2 = [0, 0, 4]$ in (2), so $[0, 0, 2]$ in (1).

Where did I make the mistake?

Answer 1

You are calculating the second cross product still in the original coordinates. You have $e_1\times e_2=e_3$, so $$ f_1\times f_2=\frac12\,f_3. $$ That last $1/2$ is the one you are missing.