Some background: I am self studying dynamics and I have encountered a fundamental problem with either my understanding of linear algebra, or I am just plain dumb. So, I print screened the page of the book we're on. Now let me try to reduce some ambiguity in my question, since my last one was not clear enough, I have a general understanding of the cross product and unit vectors. With this question it's stating the $G$ is the momentum of the particle is $G = mvn$, where $n$ is the unit vector in the direction of velocity. When we try to take the moment of the position vector from $O$ to $P$ with the momentum we get:
$$ \begin{align} M^{G/O} &= p^{P/O} \times G=p^{Q/O} \times G \tag{1} & \text{($G/O$ means vector $G$ with respect to $O$)}\\ &= y_0 n_y \times mv(\cos\theta n_x + \sin \theta n_z) \tag{2}\\ &= mvy_0(\sin \theta n_x − \cos \theta n_z) \tag{3} \end{align} $$
Questions:
A.] (I figured this one out on my own, look down for my reasoning.) How does $G = mvn$ imply that $G = mv(\cos \theta n_x + \sin \theta n_z)$? Or more specifically, how does $n = (\cos \theta n_x + \sin \theta n_z)$.
B.] How do we actually do the cross product on (2) to get to (3) if there are no explicit components of these vectors, B-1]how do you set up the determinant matrix to reach (3).
C.] How does the position vector from O -> P = the position vector from O -> Q if the particle at position P has a position in 3 dimensions, so how is it possible that $$\text{the position vector from O -> Q} = y_0n_y,$$ which is just the the distance $y_0$ in the direction of $n_y$?
My solution for the first question, A.]
Since $n$ is pointing in the same direction as velocity and momentum, n can be broken up into its respective components which are simple functions of the angle it makes with the Q axis, so we take the $\cos\theta$ and take the component that is in the $n_x$ direction. Which forms the vector component $(\cos\theta n_x)$, we then add it to the vector component in the $n_z$ direction, $(\sin\theta n_z)$, Therefore, $$ n = (\cos\theta n_x + \sin \theta n_z) $$
Here is a link to the pdf of the book if anyone is interested:
Fundamentals of Applied Dynamics - Tenenbaum
