I was given the following definition of the cross product:
The vector product $\underline{a}\times\underline{b}$ is defined as the vector with magnitude $\lvert\underline{a} \times \underline{b}\rvert = \vert\underline{a}\rvert\lvert\underline{b}\rvert \sin{\theta}$ and direction perpendicular to both $\underline{a}$ and $\underline{b}$, with $\theta$ the angle measured from $\underline{a}$ to $\underline{b}$
My understanding is that we measure angles anticlockwise by convention. And so if we were to try and compute $\underline{\hat{j}}\times\underline{\hat{i}}$ for example, the angle between these two vectors would be $\frac{3\pi}{2}$ and thus by the above definition, we have that the magnitude of $\underline{\hat{j}}\times\underline{\hat{i}}$ is $-1$ which is impossible because magnitudes of vectors must be non-negative.
I know I'm going wrong with my understanding here somewhere, I just don't understand where specifically.
(Also, I do understand right hand convention, and the fact that $\underline{\hat{j}}\times\underline{\hat{i}} = -\underline{\hat{i}}\times\underline{\hat{j}}$. However in this case, it is not that the magnitude is opposite, it's the direction which is opposite.)