Angle between two 3D Vectors

there seems to be no easy way to calculate the cross product or magnitude without having a bunch of scrap numbers in between.

Can you clarify what your concern is here? I agree that I don't see an easy way to perform the calculation without also calculating the intermediate numbers. Whether in VBA or Python or spreadsheet formula or pencil and paper and slide rule, I don't see any way to perform the overall calculation without calculating the various intermediate values that go into the calculation.

Perhaps your concern is not so much about the need to calculate the intermediate values, but that you don't want to have to look at them in the spreadsheet. You want to nest all of the intermediate calculations into one big formula that performs all of the intermediate calculations and the final calculation in one cell? It's a bit of work and you have to be real careful with parentheses and such when you nest that much into a single cell formula, but it seems like it should not be too difficult:

1) cross product i term b*z-c*y, cross product j term a*z-c*x, cross product k term a*y-b*x
2) magnitude of a vector is SQRT(SUMSQ(vector elements)) so SQRT(SUMSQ(b*z-c*y,a*z-c*x,a*y-b*x))
3) dot product is a simple SUMPRODUCT(vector elements a, vector elements b) SUMPRODUCT({a,b,c},{x,y,z}) or references to the ranges with those values.
4) finally, take the arctan of the ratio =ATAN(SQRT(SUMSQ(...))/SUMPRODUCT(...))
5) test and debug to make sure I got all the parentheses and such in the right place.

Is that the main concern -- about nesting everything into a single cell formula?