breakdown the vector into vectors perpendicular and parallel

breakdown the vector <3,-1,-2> into vectors perpendicular and parallel to <1,-2,2>

Solution

let u =<3,-1,-2> , v=<1,-2,2>

|v|=[1²+(-2)²+2²] =9

u.v =<3,-1,-2>.<1,-2,2> =(3*1)+(-1*-2)+(-2*2) =1

vectors parallel to <1,-2,2> is (u.v_/|v|²)v

vectors parallel to <1,-2,2> is (1_/|9|²)<1,-2,2>

vectors parallel to <1,-2,2> is (1_/9)<1,-2,2>

vectors parallel to <1,-2,2> is <1_/9,-2_/9,2_/9>

vectors perpendicular to <1,-2,2> is  <3,-1,-2>-<1_/9,-2_/9,2_/9>

vectors perpendicular to <1,-2,2> is  <3-(1_/9),-1+(2_/9),-2-(2_/9)>

vectors perpendicular to <1,-2,2> is  <26_/9,-7_/9,-20_/9>