In my opinion Hibbeler's book on statics (Engineering Mechanics Statics, 12th ed) is one of the most approachable on the subject. On pg.123 he defines the Vector Cross Product in its Cartesian notation as follows; $$ \vec{A}\times\vec{B}=(A_yB_z-A_zB_y)\vec{i}-(A_xB_z-A_zB_x)\vec{j}+(A_xB_y-A_yB_x)\vec{k}\\ $$ I understand the $\vec{i},\vec{j}\;{and}\;\vec{k}$ terms represent unit vectors which represent the direction of the magnitude of the $\vec{A}$ components designated by $A_x,\;A_y\;{and}\;A_z$, and the $\vec{B}$ components designated by $B_x,\;B_y\;{and}\;B_z$.
My question is, why is this relationship not defined as follows; $$ \vec{A}\times\vec{B}=(A_yB_z-A_zB_y)\vec{x}-(A_xB_z-A_zB_x)\vec{y}+(A_xB_y-A_yB_x)\vec{z}\\ $$ Replacing the $\vec{i},\vec{j}\;{and}\;\vec{k}$ terms with $\vec{x},\vec{y}\;{and}\;\vec{z}$, so as to be consistent with the notation of the vector components? What is gained by the introduction of this additional representation of these dimensions?
Thank you
If we take 3-space as embedded in the four dimensional real quaternions, as those elements having first coordinate $0$, then a typical element is $0+xi+yj+zk$ and there is a multiplication defined on the quaternions by defining $ij=k,\ jk=i,\ ki=j,$ with the other products being $ji=-k,\ kj=-i,\ ik=-j,$ then the cross product corresponds to quaternion multiplication.
Note one also defines $ii=jj=kk=0$ and extends a multiplication linearly.
Added note: The question was about why the specific letters $i,j,k$ are sometimes used for the unit vectors in the definition of cross product. What I'm getting at here is that it may be due to the facts that (1) These letters are almost universally used in describing the quaternions, and (2) the quaternion multiplication on the last three coordinates happens to correspond componentwise to the cross product.