Consider two vectors $\vec a$, $\vec b$ whose components are not known. $|\vec a| = 3$, $|\vec b| = 2$ and an angle between them is $120^\circ$. Without assuming any particular components, calculate the magnitude of the vector product $|(\vec a − 3\vec b) \times (2\vec a + \vec b)|$.
I understand that when I just want to find the vector product given the above information I would use: $$|\vec a \times \vec b| = |\vec a||\vec b|\sin(\theta).$$
When I do this I end up with $|\vec a \times \vec b| = 3\sqrt 3$. I am not sure whether I am supposed to do this and if I am where I go on from here to calculate $|(\vec a − 3\vec b) \times (2\vec a + \vec b)|$.
PS: This is my first time here so I am not sure how to add the little arrows above the terms to show they are vectors.
Note that for cross-product, we have $(\vec b × \vec a)=-(\vec a × \vec b)$.
$$ \begin{align*} & (\vec a − 3\vec b) × (2\vec a + \vec b) \\ & = (\vec a × 2\vec a) + (\vec a × \vec b) + (-3\vec b × 2\vec a) + (-3\vec b × \vec b) \\ & = (\vec a × \vec b) + (-6)\color{blue}{(\vec b × \vec a)} \\ & = (\vec a × \vec b) + (-6)\color{blue}{(-(\vec a × \vec b))} \\ & = 7(\vec a × \vec b) \end{align*} $$
Does this help?