the vector is $A=-a_yz+a_zy$ starting at the point $P_1(0,-2,3)$ and is directed towards the point $P_2(\sqrt{3},-60^{\circ},1)$.
I am a little confused by the definition of the vector. I can't find this type of representation - usually it is given by it's coordinates, or the form $A_x+B_y+C_z=D$. I don't understand the terms used in the given equation can anyone explain this to me please?
I suspect $z$ and $y$ in $A=-a_yz+a_zy$ are basis vectors. This would be the same as writing $$A=\pmatrix{0 \\ a_z \\-a_y}.$$
Writing $A=A_x+A_y+A_z$ is actually wrong, since $A_x,A_y,A_z$ are everyday numbers (scalars), while $A$ is a vector, and you can't get a vector by adding three scalars. Instead (if in Cartesian coordinates) it should be $A=A_x\hat{x}+A_y\hat{y}+A_z\hat{z},$ where that "hat" denotes that we're talking about a vector of length one (a unit vector). Thus, the equation $A=-a_yz+a_zy$ would be less confusing if written as $A=-a_y \hat{z}+a_z\hat{y}$.
The coordinates for $P_2(\sqrt{3},-60^{\circ},1)$ would only make sense if the point was given in spherical or cylindrical coordinates, although it is not standard to give the values of the entries in degrees! Without further context, I unfortunately can't say anything more about that.