[Blindmath] Vector CrossMultiplication Question

[Rick Thomas]

Hi:
I am reviewing Vector Cross Multiplication.
One article claims that step 1 is to decompose any vector into Unit Form
using (i,j,k)
Others seem to use (i,j,k) only as component place holders in their
equations unless they are assuming readers will know to do the unit
decomposition before using their formulas.
One article did not use (i,j,k) in their formula at all.
Below are 2 solution formulas:
Can you put into words whether decomposition needs to be performed to
perform cross multiplication on 2 vectors prior to using them in the formula
or show an example to explain a solution?
Note: I use * to denote multiplication.
First without (i,j,k)
If the components for vectors A and B are known, then we can express the
components of their cross product, C = A*B:
Cx = (Ay*Bz - Az*By)
Cy = (Az*Bx - Ax*Bz)
Cz = (Ax*By - Ay*Bx)
Second Article using (i,j,k)
To take the cross product of two general vectors, we first decompose the
vectors using the unit vectors i,j,k.
Then proceed to distribute the cross product across the sums, using the
rules to do the cross products between unit vectors. 
We can do this for arbitrary vectors 
u = u1, u2, u3)
 and 
v = (v1, v2, v3)
 to get a general formula: 
u = u1i + u2j + u3k
v = v1i + v2j + v3k
= 
(
u1*v2 - u2*v1)k
+
(u3*v1 - u1*v3)j
+
(u2*v3 - u3*v2)i
OK, so the above 2 methods look pretty similar but for the use of (i,j,k)
Can you clear up this confusion for me either in words or by an example of 2
vectors with component numbers not in unit form via a stepwise solution?
I have not been able to figure this out in several days of googling.
Are examples they give just specifying (i,j,k) just using them as place
holders or do they actually calculate (i,j,k) and multiply the calculations
by their values in the second example form?
Rick USA