Given two non null vectors #vec a# and #vec b# the first #vec a# allways can be decomposed as a sum of two components: one parallel to #vec b# and another perpendicular to #vec b#.
The parallel component is the projection of #vec a# onto #vec b# or
#vec a_P = << vec a, (vec b)/norm(vec b) >> (vec b)/norm(vec b)= << vec a, vec b >> (vec b)/norm(vec b)^2#
and the perpendicular component given by
#vec a_T = vec a - vec a_P = vec a - << vec a, vec b >> (vec b)/norm(vec b)^2#
So index #P# for parallel and #T# for perpendicular. We can verify that
#vec a_P +vec a_T = vec a#
#<< vec a_P, vec a_T>> = << << vec a, vec b >> (vec b)/norm(vec b)^2, vec a - << vec a, vec b >> (vec b)/norm(vec b)^2 >> = 0#
#<< vec a_T, vec b>> = << vec a - << vec a, vec b >> (vec b)/norm(vec b)^2, vec b >> = 0#
In our case
#vec a_P = 10/21{-2hat i,-hat j,4hat k}#
#vec a_T = 1/21{83 hat i,94 hat j, 65 hat k}#
