I entered a perpendicular frame using the following mathmatica code:

{perpframe[1], perpframe[2], perpframe[3]} = {{Cos[r] Cos[t] - Cos[s] Sin[r] Sin[t], Cos[s] Cos[t] Sin[r] + Cos[r] Sin[t], Sin[r] Sin[s]}, {(-Cos[t]) Sin[r] - Cos[r] Cos[s] Sin[t], Cos[r] Cos[s] Cos[t] - Sin[r] Sin[t], Cos[r] Sin[s]}, {Sin[s] Sin[t], (-Cos[t]) Sin[s], Cos[s]}}

Next I entered a cleared 3D point {x,y,z} and calculated

({x,y,z} . perpframe[1]) perpframe[1] +

({x,y,z} . perpframe[2]) perpframe[2]

+ ({x,y,z} . perpframe[3]) perpframe[3]

Last, I applied trig identities and the output was {x,y,z}

Questions:

What is the relationship between the location of {x,y,z} and the location of

({x,y,z}.perpframe[1]) perpframe[1] and also the relationship between the location of {x,y,z} and the location of

({x,y,z}.perpframe[1]) perpframe[1] + ({x,y,z}.perpframe[2]) perpframe[2]?

Is it just that it places it on the perpendicular plane?