well you start with the scalar dot product
vec A * vec B = abs vec A abs vec B cos phi, where phi is the angle between the vectors
then you insist that vec B be the unit vector hat B = (vec B)/(abs vec B) so that
abs hat B = 1
and
vec A * hat B = abs vec A (1) cos phi = abs vec A cos phi which is the length of vec A in the \hat B direction
so that we can actually say that
(vec A * (vec B)/(abs vec B) = vec A * hat B = abs vec A cos phi )\ hat B
so with vec A = <4,-1,6 >
and vec B = <1,3,-2 > meaning hat B = 1/sqrt(14) <1,3,-2 >
(vec A * hat B = 1/sqrt(14) ( <4,-1,6 > * <1,3,-2 > ) )hat B
= 1/sqrt(14) (4 - 3 - 12 ) hat B
= - (11)/sqrt(14) hat B
