Given two non null vectors vec a→a and vec b→b the first vec a→a allways can be decomposed as a sum of two components: one parallel to vec b→b and another perpendicular to vec b→b.
The parallel component is the projection of vec a→a onto vec b→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→aP=⟨→a,→b∥∥∥→b∥∥∥⟩→b∥∥∥→b∥∥∥=⟨→a,→b⟩→b∥∥∥→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→aT=→a−→aP=→a−⟨→a,→b⟩→b∥∥∥→b∥∥∥2
So index PP for parallel and TT for perpendicular. We can verify that
vec a_P +vec a_T = vec a→aP+→aT=→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⟨→aP,→aT⟩=⟨⟨→a,→b⟩→b∥∥∥→b∥∥∥2,→a−⟨→a,→b⟩→b∥∥∥→b∥∥∥2⟩=0
<< vec a_T, vec b>> = << vec a - << vec a, vec b >> (vec b)/norm(vec b)^2, vec b >> = 0⟨→aT,→b⟩=⟨→a−⟨→a,→b⟩→b∥∥∥→b∥∥∥2,→b⟩=0
In our case
vec a_P = 10/21{-2hat i,-hat j,4hat k}→aP=1021{−2ˆi,−ˆj,4ˆk}
vec a_T = 1/21{83 hat i,94 hat j, 65 hat k}→aT=121{83ˆi,94ˆj,65ˆk}
