If a, b, c are three mutually perpendicular vectors, then the projection of the vector [l a/|a| + m b/|b| + n (a+b)/|a+b|] along the angular bisector of vectors a and b may be given as

a) (l² + m²)/(l²+m²+n²)^1/2

b) (l²+m²+n²)^1/2

c) (l²+m²)^1/2/(l²+m²+n²)^1/2

d) (l+m)/2^1/2

L(i/1)+M(j/1) +N(i+j)/root 2.

Now the angular bisector of a and b is given by a/mod(a) +b/mod(b)

implies i+j

now projection of the 1st vector on i+j  is {L(i/1)+M(j/1) +N(i+j)/root 2}cos(angle between the vectors).

See if the method works.

d) (l+m)/2^1/2

easy

use i,j,k as translational axes