b=(5,2,1) ==> λb=(5λ+2λ+λ) ==> |λb| =λ√30

a=(1,1,1) ==> |a| =√(1+1+1)=√3

a. λb=(5 λ,2 λ, λ).(1,1,1) =(5 λ+2 λ+ λ) =8λ

cos β = a. λb/ |a| |λb|

cos β =8λ/λ√30*√3 = 8/√90

cos β = |Proj λb sobre a | / λ√30

|Proj λb sobre a | = cos β * λ√30 = 8/√90 *λ√30 =8λ/√3

vetor unitário a ==> (1/√3,1/√3,1/√3)

8λ/√3 * (1/√3,1/√3,1/√3) = (1,1,1)

8λ/√3 * 1/√3 =1 ==> 8λ/3=1 ==>λ =3/8