I have two 3D-vectors. A Force vector F = (1, -2, 2) and an arm vector r = (-1, 5, 3). If I calculate the moment with the cross product, I get cross product (-16, -5, 3) and resulting moment of sqrt((-16)^2 + (-5)^2 + 3^2) = sqrt(290). sqrt(290) is about 17.03 Nm.
If I calculate the moment with M = |F| x |r| with |F| = 3 and |r| = sqrt(35) I get 3*sqrt(35) which is about 17.75 Nm.
My question is: How can the difference in moments be explained? Both methods to calculate the moment are valid I think.
By definition of cross-product, $$\underline{F}\times\underline{r}=|\underline{F}||\underline{r}|\sin \theta\underline{\hat{n}}$$ $\underline{\hat{n}}$ is a unit vector perpendicular to $\underline{F}$ and $\underline{r}$.
This cannot have the same magnitude as $$|\underline{F}||\underline{r}|$$ unless $\sin\theta=1$, i.e. unless $\underline{F}\perp\underline{r}$